QDM Lattice 4x2
Contents
[1]:
import numpy as np
import pandas as pd
from scipy.linalg import expm, null_space, solve
from scipy.sparse import csr_matrix
from scipy.sparse.linalg import expm_multiply, eigsh
import matplotlib.pyplot as plt
import networkx as nx
from networkx.algorithms.isomorphism import GraphMatcher
import pyvis.network as nt
import igraph
from qlinks.lattice.square_lattice import SquareLattice
from qlinks.symmetry.gauss_law import GaussLaw
from qlinks.solver.deep_first_search import DeepFirstSearch
from qlinks.computation_basis import ComputationBasis
from qlinks.model import QuantumLinkModel
from qlinks.visualizer.graph import GraphVisualizer
pd.set_option('display.max_rows', None)
[2]:
def format_custom_index(index):
return [f"({i}) {idx}" for i, idx in enumerate(index)]
[3]:
def setup_dimer_model(lattice_shape, n_solution, coup_j, coup_rk):
gauss_law = GaussLaw.from_staggered_charge_distri(*lattice_shape)
gauss_law.flux_sector = (0, 0)
dfs = DeepFirstSearch(gauss_law, max_steps=int(1e+8))
basis = gauss_law.to_basis(dfs.solve(n_solution))
model = QuantumLinkModel(coup_j, coup_rk, lattice_shape, basis)
return basis, model
[4]:
def setup_link_model(lattice_shape, n_solution, coup_j, coup_rk):
gauss_law = GaussLaw.from_zero_charge_distri(*lattice_shape)
gauss_law.flux_sector = (0, 0)
dfs = DeepFirstSearch(gauss_law, max_steps=int(1e+8))
basis = gauss_law.to_basis(dfs.solve(n_solution))
model = QuantumLinkModel(coup_j, coup_rk, lattice_shape, basis)
return basis, model
QDM Lattice 4x2#
[9]:
coup_j, coup_rk = (1, 1)
basis, model = setup_dimer_model(lattice_shape=(4, 2), n_solution=16, coup_j=coup_j, coup_rk=coup_rk)
evals, evecs = np.linalg.eigh(model.hamiltonian)
2023-12-23 23:45:25 [deep_first_search.py] INFO: Deep First Search starts.
2023-12-23 23:45:25 [deep_first_search.py] INFO: Found 16 Solutions as required in 103 steps.
[10]:
df = basis.dataframe.copy(deep=True)
df.index = format_custom_index(df.index)
df
[10]:
| 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| (0) 17595 | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 0 | 1 | 1 | 1 | 0 | 1 | 1 |
| (1) 18867 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | 1 | 1 | 0 | 0 | 1 | 1 |
| (2) 19638 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 0 | 1 | 1 | 0 | 1 | 1 | 0 |
| (3) 36135 | 1 | 0 | 0 | 0 | 1 | 1 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 1 |
| (4) 37947 | 1 | 0 | 0 | 1 | 0 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 0 | 1 | 1 |
| (5) 39219 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 |
| (6) 39990 | 1 | 0 | 0 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 0 | 1 | 1 | 0 |
| (7) 44307 | 1 | 0 | 1 | 0 | 1 | 1 | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 1 |
| (8) 50283 | 1 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 1 | 0 | 1 | 0 | 1 | 1 |
| (9) 51555 | 1 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 0 | 1 | 1 | 0 | 0 | 0 | 1 | 1 |
| (10) 52326 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 |
| (11) 52773 | 1 | 1 | 0 | 0 | 1 | 1 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 0 | 1 |
| (12) 55410 | 1 | 1 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 0 | 0 | 1 | 0 |
| (13) 55857 | 1 | 1 | 0 | 1 | 1 | 0 | 1 | 0 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 1 |
| (14) 60498 | 1 | 1 | 1 | 0 | 1 | 1 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 0 | 1 | 0 |
| (15) 60945 | 1 | 1 | 1 | 0 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 |
[11]:
evecs_df = pd.DataFrame.from_dict(
{
"eval": evals,
"kin": [(evec.T @ model.kinetic_term @ evec).item() for evec in evecs.T],
"pot": [(evec.T @ model.potential_term @ evec).item() for evec in evecs.T],
}
)
evecs_df[evecs_df["kin"].abs() < 1e-12]
[11]:
| eval | kin | pot | |
|---|---|---|---|
| 7 | 3.0 | 3.451266e-31 | 3.0 |
| 8 | 3.0 | 0.000000e+00 | 3.0 |
| 9 | 3.0 | 1.577722e-30 | 3.0 |
[12]:
k_rank = np.linalg.matrix_rank(model.kinetic_term)
h_rank = np.linalg.matrix_rank(model.hamiltonian)
print(k_rank, h_rank, basis.n_states, (basis.n_states - k_rank) / basis.n_states)
10 15 16 0.375
[13]:
g = nx.from_numpy_array(-model.kinetic_term)
# assert nx.is_bipartite(g)
# left, right = nx.bipartite.sets(g)
# pos = nx.bipartite_layout(g, left, align="horizontal")
# pos = nx.spectral_layout(g)
# pos = {k: v + np.random.normal(0, 0.05, 2) for k, v in pos.items()}
# nx.draw(g, pos, with_labels=True, arrows=True, connectionstyle="arc3,rad=0.2")
# plt.show()
plt.spy(evecs, precision=1e-12, markersize=12)
plt.show()
[124]:
G_pattern = nx.Graph()
G_pattern.add_nodes_from([0, 1, 2, 3, 4, 5, 6, 7, 8])
G_pattern.add_edges_from([(0, 1), (0, 2), (1, 3), (1, 4), (2, 4), (2, 5), (3, 6), (4, 6), (4, 7), (5, 7), (6, 8), (7, 8)])
# nx.draw(G_pattern, nx.spectral_layout(G_pattern), with_labels=True)
gm = GraphMatcher(g, G_pattern)
subgraphs = set([frozenset(subgraph_nodes) for subgraph_nodes in gm.subgraph_isomorphisms_iter()])
print(subgraphs)
{frozenset({3, 5, 7, 10, 11, 12, 13, 14, 15}), frozenset({0, 1, 2, 4, 5, 6, 8, 9, 10})}
[10]:
net = nt.Network(notebook=True, filter_menu=True, select_menu=True)
net.from_nx(g)
net.show_buttons(filter_=['nodes', 'edges', 'physics'])
net.show("nx.html")
Warning: When cdn_resources is 'local' jupyter notebook has issues displaying graphics on chrome/safari. Use cdn_resources='in_line' or cdn_resources='remote' if you have issues viewing graphics in a notebook.
nx.html
[10]:
[13]:
ig = igraph.Graph.from_networkx(g)
ig.vs["label"] = [str(i) for i in range(ig.vcount())]
# color = ["orange" for i in range(ig.vcount())]
# highlight = [
# [14, 13, 11, 7],
# [1, 8, 2, 4],
# ]
# highlight_color = ["aqua", "yellowgreen"]
# for i, nodes in enumerate(highlight):
# for node in nodes:
# color[node] = highlight_color[i]
# ig.vs["color"] = color
# fig, ax = plt.subplots(figsize=(6, 6), facecolor="white")
igraph.plot(
ig,
layout=ig.layout_kamada_kawai(),
vertex_size=16,
vertex_color="orange",
# target=ax
)
[13]:
[14]:
entropies = np.asarray([model.entropy(evecs[:, i], 1, 0) for i in range(basis.n_states)])
entropy_df = pd.Series(entropies, name="entropy")
entropy_df
[14]:
0 1.536650
1 1.899841
2 0.695771
3 1.562885
4 1.528069
5 1.600664
6 1.932517
7 1.525501
8 1.611432
9 1.481633
10 1.941802
11 0.973698
12 1.147446
13 1.309708
14 1.502661
15 1.501048
Name: entropy, dtype: float64
[15]:
entropy_df.plot(style='o')
[15]:
<Axes: >
[17]:
i = 8
plt.plot(evecs[:, i], linestyle='--', marker='o')
np.where(np.abs(evecs[:, i]) > 0.1)[0]
[17]:
array([ 1, 2, 4, 7, 8, 11, 13, 14])
[19]:
fig, axes = plt.subplots(4, 4, figsize=(16, 16), facecolor="white")
ax = axes.flatten()
for i in range(basis.n_states):
g = GraphVisualizer(SquareLattice(*model.shape, basis.dataframe.iloc[i]))
g.plot(show=False, ax=ax[i], node_size=800)
ax[i].set_title(f"{df.index[i]}")
[21]:
coup_j = np.ones((np.prod(model.shape), 1))
coup_j[0] = 1.5
energy_lump_model = QuantumLinkModel(coup_j, coup_rk, model.shape, basis)
energy_lump_ham = energy_lump_model.hamiltonian
scar_state = np.zeros((basis.n_states, 1))
scar_state[np.array([1, 2, 4, 8]), 0] = np.array([1, 1, -1, -1])
scar_state /= np.linalg.norm(scar_state)
start, stop, num = 0, 1000, 100
evol_states = expm_multiply(-1j * energy_lump_ham, scar_state, start=start, stop=stop, num=num)
fidelity = [np.abs(evol_state.T @ scar_state).item() ** 2 for evol_state in evol_states]
entropy = [model.entropy(evol_state.flatten(), 1, 0) for evol_state in evol_states]
plt.plot(np.linspace(start, stop, num), fidelity, linestyle="--", marker="o")
plt.ylim(0.0001, 1.1)
# plt.yscale('log')
plt.xlabel("t")
plt.ylabel("Fidelity(t)")
plt.tight_layout()
plt.show()
plt.plot(np.linspace(start, stop, num), np.round(entropy, 12), linestyle="--", marker="o")
plt.xlabel("t")
plt.ylabel("EE(t)")
plt.tight_layout()
plt.show()
final_state = np.real_if_close(evol_states[-1], tol=1e-12)
print(f"final fidelity = {fidelity[-1]}")
print(f"O_kin @ psi_scar = {(model.kinetic_term @ final_state).flatten()}")
print(f"<O_kin> = {(final_state.T @ model.kinetic_term @ final_state).item()}")
print(f"<O_pot> = {(final_state.T @ model.potential_term @ final_state).item()}")
print(f"<H> = {(final_state.T @ model.hamiltonian @ final_state).item()}")
final fidelity = 0.2163724158208147
O_kin @ psi_scar = [-0.244725 -0.2726498j 0.13566086-0.01666943j -0.05689253-0.01094762j
0.16312646+0.07970323j -0.08939803+0.06206632j 0.1732982 +0.11400362j
0.02198015-0.14927196j -0.00661834+0.06313454j 0.16816636-0.08968338j
-0.26670515-0.12337784j -0.09177027-0.22724696j 0.02588716-0.0098794j
0.16312646+0.07970323j -0.00661834+0.06313454j 0.02588716-0.0098794j
0.32625293+0.15940645j]
<O_kin> = (0.12547370021383802-0.04254736654748517j)
<O_pot> = (0.7340468683168595+2.010551039444004j)
<H> = (0.8595205685306977+1.9680036728965187j)
QDM Lattice 4x4#
[22]:
coup_j, coup_rk = (1, 1)
basis, model = setup_dimer_model(lattice_shape=(4, 4), n_solution=132, coup_j=coup_j, coup_rk=coup_rk)
evals, evecs = np.linalg.eigh(model.hamiltonian)
2023-12-23 23:48:46 [deep_first_search.py] INFO: Deep First Search starts.
2023-12-23 23:48:46 [deep_first_search.py] INFO: Found 132 Solutions as required in 1626 steps.
[23]:
df = basis.dataframe.copy(deep=True)
df.index = format_custom_index(df.index)
df
[23]:
| 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | ... | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 30 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| (0) 1142025915 | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | ... | 1 | 0 | 1 | 0 | 1 | 1 | 1 | 0 | 1 | 1 |
| (1) 1143328443 | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | ... | 1 | 0 | 1 | 0 | 1 | 1 | 1 | 0 | 1 | 1 |
| (2) 1146285243 | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | ... | 0 | 0 | 1 | 0 | 1 | 1 | 1 | 0 | 1 | 1 |
| (3) 1147587771 | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | ... | 0 | 0 | 1 | 0 | 1 | 1 | 1 | 0 | 1 | 1 |
| (4) 1147913403 | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | ... | 0 | 0 | 1 | 0 | 1 | 1 | 1 | 0 | 1 | 1 |
| (5) 1147914675 | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | ... | 0 | 1 | 1 | 0 | 1 | 1 | 0 | 0 | 1 | 1 |
| (6) 1152797883 | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 0 | ... | 0 | 0 | 1 | 0 | 1 | 1 | 1 | 0 | 1 | 1 |
| (7) 1152818235 | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 0 | ... | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 0 | 1 | 1 |
| (8) 1153123515 | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 0 | ... | 0 | 0 | 1 | 0 | 1 | 1 | 1 | 0 | 1 | 1 |
| (9) 1153124787 | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 0 | ... | 0 | 1 | 1 | 0 | 1 | 1 | 0 | 0 | 1 | 1 |
| (10) 1153143867 | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 0 | ... | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 0 | 1 | 1 |
| (11) 1153145139 | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 0 | ... | 0 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 |
| (12) 1153150227 | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 0 | ... | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 1 |
| (13) 1153161777 | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 0 | ... | 1 | 0 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 1 |
| (14) 1153166865 | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 0 | ... | 1 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 |
| (15) 1231275195 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 0 | 1 | ... | 0 | 0 | 1 | 0 | 1 | 1 | 1 | 0 | 1 | 1 |
| (16) 1231276467 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 0 | 1 | ... | 0 | 1 | 1 | 0 | 1 | 1 | 0 | 0 | 1 | 1 |
| (17) 1236485307 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | ... | 0 | 0 | 1 | 0 | 1 | 1 | 1 | 0 | 1 | 1 |
| (18) 1236486579 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | ... | 0 | 1 | 1 | 0 | 1 | 1 | 0 | 0 | 1 | 1 |
| (19) 1236505659 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | ... | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 0 | 1 | 1 |
| (20) 1236506931 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | ... | 0 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 |
| (21) 1236512019 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | ... | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 1 |
| (22) 1236523569 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | ... | 1 | 0 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 1 |
| (23) 1236528657 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | ... | 1 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 |
| (24) 1276243638 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | ... | 1 | 0 | 1 | 0 | 1 | 1 | 0 | 1 | 1 | 0 |
| (25) 1277546166 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | ... | 1 | 0 | 1 | 0 | 1 | 1 | 0 | 1 | 1 | 0 |
| (26) 1280502966 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 1 | ... | 0 | 0 | 1 | 0 | 1 | 1 | 0 | 1 | 1 | 0 |
| (27) 1281805494 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 1 | ... | 0 | 0 | 1 | 0 | 1 | 1 | 0 | 1 | 1 | 0 |
| (28) 1282131126 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 1 | ... | 0 | 0 | 1 | 0 | 1 | 1 | 0 | 1 | 1 | 0 |
| (29) 1287015606 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 0 | ... | 0 | 0 | 1 | 0 | 1 | 1 | 0 | 1 | 1 | 0 |
| (30) 1287035958 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 0 | ... | 0 | 0 | 0 | 0 | 1 | 1 | 0 | 1 | 1 | 0 |
| (31) 1287341238 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 0 | ... | 0 | 0 | 1 | 0 | 1 | 1 | 0 | 1 | 1 | 0 |
| (32) 1287361590 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 0 | ... | 0 | 0 | 0 | 0 | 1 | 1 | 0 | 1 | 1 | 0 |
| (33) 2366876967 | 1 | 0 | 0 | 0 | 1 | 1 | 0 | 1 | 0 | 0 | ... | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 1 |
| (34) 2366893605 | 1 | 0 | 0 | 0 | 1 | 1 | 0 | 1 | 0 | 0 | ... | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 0 | 1 |
| (35) 2368179495 | 1 | 0 | 0 | 0 | 1 | 1 | 0 | 1 | 0 | 0 | ... | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 1 |
| (36) 2368196133 | 1 | 0 | 0 | 0 | 1 | 1 | 0 | 1 | 0 | 0 | ... | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 0 | 1 |
| (37) 2486586555 | 1 | 0 | 0 | 1 | 0 | 1 | 0 | 0 | 0 | 0 | ... | 0 | 0 | 1 | 0 | 1 | 1 | 1 | 0 | 1 | 1 |
| (38) 2486606907 | 1 | 0 | 0 | 1 | 0 | 1 | 0 | 0 | 0 | 0 | ... | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 0 | 1 | 1 |
| (39) 2486912187 | 1 | 0 | 0 | 1 | 0 | 1 | 0 | 0 | 0 | 0 | ... | 0 | 0 | 1 | 0 | 1 | 1 | 1 | 0 | 1 | 1 |
| (40) 2486913459 | 1 | 0 | 0 | 1 | 0 | 1 | 0 | 0 | 0 | 0 | ... | 0 | 1 | 1 | 0 | 1 | 1 | 0 | 0 | 1 | 1 |
| (41) 2486932539 | 1 | 0 | 0 | 1 | 0 | 1 | 0 | 0 | 0 | 0 | ... | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 0 | 1 | 1 |
| (42) 2486933811 | 1 | 0 | 0 | 1 | 0 | 1 | 0 | 0 | 0 | 0 | ... | 0 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 |
| (43) 2486938899 | 1 | 0 | 0 | 1 | 0 | 1 | 0 | 0 | 0 | 0 | ... | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 1 |
| (44) 2486950449 | 1 | 0 | 0 | 1 | 0 | 1 | 0 | 0 | 0 | 0 | ... | 1 | 0 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 1 |
| (45) 2486955537 | 1 | 0 | 0 | 1 | 0 | 1 | 0 | 0 | 0 | 0 | ... | 1 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 |
| (46) 2570273979 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 0 | 0 | ... | 0 | 0 | 1 | 0 | 1 | 1 | 1 | 0 | 1 | 1 |
| (47) 2570275251 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 0 | 0 | ... | 0 | 1 | 1 | 0 | 1 | 1 | 0 | 0 | 1 | 1 |
| (48) 2570294331 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 0 | 0 | ... | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 0 | 1 | 1 |
| (49) 2570295603 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 0 | 0 | ... | 0 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 |
| (50) 2570300691 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 0 | 0 | ... | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 1 |
| (51) 2570312241 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 0 | 0 | ... | 1 | 0 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 1 |
| (52) 2570317329 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 0 | 0 | ... | 1 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 |
| (53) 2620804278 | 1 | 0 | 0 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | ... | 0 | 0 | 1 | 0 | 1 | 1 | 0 | 1 | 1 | 0 |
| (54) 2620824630 | 1 | 0 | 0 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | ... | 0 | 0 | 0 | 0 | 1 | 1 | 0 | 1 | 1 | 0 |
| (55) 2621129910 | 1 | 0 | 0 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | ... | 0 | 0 | 1 | 0 | 1 | 1 | 0 | 1 | 1 | 0 |
| (56) 2621150262 | 1 | 0 | 0 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | ... | 0 | 0 | 0 | 0 | 1 | 1 | 0 | 1 | 1 | 0 |
| (57) 2903721147 | 1 | 0 | 1 | 0 | 1 | 1 | 0 | 1 | 0 | 0 | ... | 0 | 0 | 1 | 0 | 1 | 1 | 1 | 0 | 1 | 1 |
| (58) 2903722419 | 1 | 0 | 1 | 0 | 1 | 1 | 0 | 1 | 0 | 0 | ... | 0 | 1 | 1 | 0 | 1 | 1 | 0 | 0 | 1 | 1 |
| (59) 2903741499 | 1 | 0 | 1 | 0 | 1 | 1 | 0 | 1 | 0 | 0 | ... | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 0 | 1 | 1 |
| (60) 2903742771 | 1 | 0 | 1 | 0 | 1 | 1 | 0 | 1 | 0 | 0 | ... | 0 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 |
| (61) 2903747859 | 1 | 0 | 1 | 0 | 1 | 1 | 0 | 1 | 0 | 0 | ... | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 1 |
| (62) 2903759409 | 1 | 0 | 1 | 0 | 1 | 1 | 0 | 1 | 0 | 0 | ... | 1 | 0 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 1 |
| (63) 2903764497 | 1 | 0 | 1 | 0 | 1 | 1 | 0 | 1 | 0 | 0 | ... | 1 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 |
| (64) 2905050387 | 1 | 0 | 1 | 0 | 1 | 1 | 0 | 1 | 0 | 0 | ... | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 1 |
| (65) 2905067025 | 1 | 0 | 1 | 0 | 1 | 1 | 0 | 1 | 0 | 0 | ... | 1 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 |
| (66) 3289509483 | 1 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | ... | 1 | 0 | 0 | 1 | 1 | 0 | 1 | 0 | 1 | 1 |
| (67) 3290812011 | 1 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | ... | 1 | 0 | 0 | 1 | 1 | 0 | 1 | 0 | 1 | 1 |
| (68) 3293768811 | 1 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | ... | 0 | 0 | 0 | 1 | 1 | 0 | 1 | 0 | 1 | 1 |
| (69) 3295071339 | 1 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | ... | 0 | 0 | 0 | 1 | 1 | 0 | 1 | 0 | 1 | 1 |
| (70) 3295396971 | 1 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | ... | 0 | 0 | 0 | 1 | 1 | 0 | 1 | 0 | 1 | 1 |
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132 rows × 32 columns
[24]:
evecs_df = pd.DataFrame.from_dict(
{
"eval": evals,
"kin": [(evec.T @ model.kinetic_term @ evec).item() for evec in evecs.T],
"pot": [(evec.T @ model.potential_term @ evec).item() for evec in evecs.T],
}
)
evecs_df[evecs_df["kin"].abs() < 1e-12]
[24]:
| eval | kin | pot | |
|---|---|---|---|
| 47 | 4.0 | -9.907176e-30 | 4.0 |
| 48 | 4.0 | -8.923989e-30 | 4.0 |
| 49 | 4.0 | -9.959369e-30 | 4.0 |
| 50 | 4.0 | -6.914859e-30 | 4.0 |
| 51 | 4.0 | -5.756219e-30 | 4.0 |
| 52 | 4.0 | 3.056836e-30 | 4.0 |
| 53 | 4.0 | -2.465190e-30 | 4.0 |
| 54 | 4.0 | -1.015658e-29 | 4.0 |
| 55 | 4.0 | -1.005798e-29 | 4.0 |
[25]:
k_rank = np.linalg.matrix_rank(model.kinetic_term)
h_rank = np.linalg.matrix_rank(model.hamiltonian)
print(k_rank, h_rank, basis.n_states, (basis.n_states - k_rank) / basis.n_states)
102 131 132 0.22727272727272727
[26]:
g = nx.from_numpy_array(-model.kinetic_term)
# left, right = nx.bipartite.sets(g)
# pos = nx.bipartite_layout(g, left, align='horizontal')
# pos = nx.spectral_layout(g)
# pos = {k: v + np.random.normal(0, 0.01, 2) for k, v in pos.items()}
# fig = plt.figure(figsize=(82, 32))
# nx.draw(g, pos, with_labels=True, node_size=5000, font_size=42, arrows=True, connectionstyle="arc3,rad=0.15")
# plt.show()
plt.spy(evecs, precision=1e-12, markersize=0.1)
plt.show()
[167]:
# G_pattern = nx.Graph()
# G_pattern.add_nodes_from([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14])
# G_pattern.add_edges_from([(0, 1), (0, 5), (2, 1), (2, 7), (1, 3), (1, 4), (3, 5), (3, 6), (4, 6), (4, 7), (5, 10), (5, 9), (6, 8), (6, 9), (7, 8), (7, 12), (8, 11), (9, 11), (10, 11), (12, 11), (3, 13), (13, 9), (4, 14), (14, 8)])
G_pattern = g.subgraph([131, 94, 116, 130, 129, 93, 92, 115, 114, 85, 84, 91, 128, 83, 82, 113])
# print(np.linalg.matrix_rank(nx.to_numpy_array(G_pattern)))
# left, right = nx.bipartite.sets(G_pattern)
nx.draw(G_pattern, nx.spectral_layout(G_pattern), with_labels=True)
gm = GraphMatcher(g, G_pattern)
subgraphs = set([frozenset(subgraph_nodes) for subgraph_nodes in gm.subgraph_isomorphisms_iter()])
print(len(subgraphs))
for subgraph in subgraphs:
if any(head_node in list(subgraph) for head_node in [131, 0, 54, 76, 97, 35, 14, 117]):
pass
else:
print(subgraph)
12
frozenset({3, 4, 69, 70, 6, 8, 72, 73, 82, 83, 84, 85, 27, 28, 29, 31})
frozenset({102, 103, 104, 105, 49, 50, 51, 52, 120, 121, 122, 123, 60, 61, 62, 63})
frozenset({128, 79, 80, 81, 82, 87, 89, 90, 91, 109, 110, 112, 113, 123, 125, 127})
frozenset({8, 9, 10, 11, 17, 18, 19, 20, 39, 40, 41, 42, 46, 47, 48, 49})
[52]:
net = nt.Network(notebook=True, filter_menu=True, select_menu=True, font_color="black")
net.from_nx(g)
for node in net.nodes:
node["title"] = str(node["id"])
net.show_buttons(filter_=['nodes', 'edges', 'physics'])
net.show("nx.html")
Warning: When cdn_resources is 'local' jupyter notebook has issues displaying graphics on chrome/safari. Use cdn_resources='in_line' or cdn_resources='remote' if you have issues viewing graphics in a notebook.
nx.html
[52]:
[31]:
ig = igraph.Graph.from_networkx(g)
ig.vs["label"] = [str(i) for i in range(ig.vcount())]
# color = ["orange" for i in range(ig.vcount())]
# highlight = [
# [34, 65, 86, 124],
# [96, 108, 107, 126],
# [93, 115, 92, 114],
# [77, 74, 15, 5],
# [59, 101, 58, 100],
# [25, 67, 26, 68],
# [55, 37, 32, 7],
# [43, 21, 44, 22]
# ]
# highlight_color = ["aqua", "deepskyblue", "yellowgreen", "royalblue", "blueviolet", "cornflowerblue", "limegreen", "pink"]
# for i, nodes in enumerate(highlight):
# for node in nodes:
# color[node] = highlight_color[i]
# ig.vs["color"] = color
# fig, ax = plt.subplots(figsize=(10, 10), facecolor="white")
igraph.plot(
ig,
layout=ig.layout_kamada_kawai(),
vertex_size=12,
vertex_color="orange",
vertex_label_size=8,
# target=ax
)
[31]:
[32]:
entropies = np.asarray([model.entropy(evecs[:, i], 2, 1) for i in range(basis.n_states)])
entropy_df = pd.Series(entropies, name="entropy")
entropy_df[entropy_df < 1.9]
[32]:
38 1.863804
41 1.898850
51 1.872677
53 1.834227
54 1.804352
55 1.807246
112 1.827273
123 1.881107
Name: entropy, dtype: float64
[33]:
entropy_df.plot(style='o')
[33]:
<Axes: >
[35]:
i = 50
plt.plot(evecs[:, i], linestyle='--', marker='o')
np.where(np.abs(evecs[:, i]) > 0.2)[0]
[35]:
array([ 25, 26, 67, 68, 96, 107, 108, 126])
[36]:
# {i: len(np.where(np.abs(evecs[:, i]) > 1e-12)[0]) for i in range(300)}
fig, axes = plt.subplots(2, 2, figsize=(16, 12), facecolor="white")
ax = axes.flatten()
for i, val in enumerate([25, 26, 67, 68]):
g = GraphVisualizer(SquareLattice(*model.shape, basis.dataframe.iloc[val]))
g.plot(show=False, ax=ax[i], node_size=800)
ax[i].set_title(f"{df.index[val]}")
[40]:
coup_j = np.ones((np.prod(model.shape), 1))
coup_j[5] = 1.1
energy_lump_model = QuantumLinkModel(coup_j, coup_rk, model.shape, basis)
energy_lump_ham = energy_lump_model.hamiltonian
scar_state = np.zeros((basis.n_states, 1))
scar_state[np.array([25, 26, 67, 68]), 0] = np.array([1, -1, -1, 1])
scar_state /= np.linalg.norm(scar_state)
start, stop, num = 0, 200, 100
evol_states = expm_multiply(-1j * energy_lump_ham, scar_state, start=start, stop=stop, num=num)
fidelity = [np.abs(evol_state.T @ scar_state).item() ** 2 for evol_state in evol_states]
entropy = [model.entropy(evol_state.flatten(), 2, 0) for evol_state in evol_states]
plt.plot(np.linspace(start, stop, num), fidelity, linestyle="--", marker="o")
plt.ylim(0.0001, 1.1)
# plt.yscale('log')
plt.xlabel("t")
plt.ylabel("Fidelity(t)")
plt.tight_layout()
plt.show()
plt.plot(np.linspace(start, stop, num), np.round(entropy, 12), linestyle="--", marker="o")
plt.xlabel("t")
plt.ylabel("EE(t)")
plt.tight_layout()
plt.show()
final_state = np.real_if_close(evol_states[-1], tol=1e-12)
print(f"final fidelity = {fidelity[-1]}")
print(f"O_kin @ psi_scar = {(model.kinetic_term @ final_state).flatten()}")
print(f"<O_kin> = {(final_state.T @ model.kinetic_term @ final_state).item()}")
print(f"<O_pot> = {(final_state.T @ model.potential_term @ final_state).item()}")
print(f"<H> = {(final_state.T @ model.hamiltonian @ final_state).item()}")
final fidelity = 0.9879392050588764
O_kin @ psi_scar = [-1.90812800e-16+8.28091669e-17j -3.65133183e-16+1.00329721e-15j
-2.35776161e-16+4.67119381e-16j -7.80625564e-18+4.16333634e-17j
-1.74795127e-02-9.03285214e-03j -2.73938464e-03+5.20666229e-03j
1.74795127e-02+9.03285214e-03j 2.73938464e-03-5.20666229e-03j
-3.22658567e-16+4.72712147e-17j -3.61119152e-03+6.08438797e-03j
3.61119152e-03-6.08438797e-03j -2.89698820e-16-2.24646690e-16j
3.85072889e-17-4.05411804e-16j -4.56055529e-17-8.64761428e-17j
-2.22121645e-16-2.27589252e-16j -2.73938464e-03+5.20666229e-03j
1.20663790e-02-7.38583877e-03j -3.61119152e-03+6.08438797e-03j
-4.42050954e-04-4.97435852e-03j -8.76035355e-17-4.77048956e-17j
-2.18963565e-03-5.76980809e-03j 5.35393878e-03-1.66115198e-03j
5.35393878e-03-1.66115198e-03j -2.48235535e-03-7.62830643e-03j
3.91205599e-02+1.00014163e-03j -2.45225784e-03-5.36830970e-03j
-2.45225784e-03-5.36830970e-03j -1.81588903e-02+1.32788326e-02j
-1.15137815e-02-1.60958899e-02j -1.52035833e-03+1.38019956e-02j
-6.87649577e-03+3.88407163e-03j -3.55428138e-02-4.03318656e-03j
-8.13810349e-03+4.73364612e-03j 6.22652007e-16-3.35461204e-18j
1.36900345e-17+2.68423982e-16j 1.23423291e-16-3.71341662e-17j
1.26827388e-16+1.89787073e-16j 2.73938464e-03-5.20666229e-03j
-1.20663790e-02+7.38583877e-03j 3.61119152e-03-6.08438797e-03j
-5.81132364e-17-1.56125113e-16j 4.42050954e-04+4.97435852e-03j
2.18963565e-03+5.76980809e-03j -5.35393878e-03+1.66115198e-03j
-5.35393878e-03+1.66115198e-03j 2.48235535e-03+7.62830643e-03j
-2.87096735e-16-2.51534904e-16j -2.18963565e-03-5.76980809e-03j
2.18963565e-03+5.76980809e-03j 1.34007389e-16-2.28116137e-16j
2.75547687e-16-3.17039042e-16j 4.45518215e-16+1.96542943e-16j
1.43290475e-16-2.00399236e-16j -6.87649577e-03+3.88407163e-03j
-5.44204673e-04+1.14773075e-03j -8.13810349e-03+4.73364612e-03j
2.35990715e-02-1.67861370e-02j 7.02563008e-17-4.23706209e-16j
5.35393878e-03-1.66115198e-03j -5.35393878e-03+1.66115198e-03j
3.03576608e-16-3.64725611e-16j 5.39771104e-16+2.56207448e-17j
1.80902577e-16-6.46329022e-17j -8.02520617e-17-6.93554591e-17j
-3.40951184e-16-5.82888883e-16j -7.08834627e-17-3.55703350e-16j
-3.91205599e-02-1.00014163e-03j 2.45225784e-03+5.36830970e-03j
2.45225784e-03+5.36830970e-03j 1.81588903e-02-1.32788326e-02j
1.52035833e-03-1.38019956e-02j 6.87649577e-03-3.88407163e-03j
1.15137815e-02+1.60958899e-02j 3.55428138e-02+4.03318656e-03j
8.13810349e-03-4.73364612e-03j 6.87649577e-03-3.88407163e-03j
5.44204673e-04-1.14773075e-03j 8.13810349e-03-4.73364612e-03j
-2.35990715e-02+1.67861370e-02j 3.67784958e-16+6.57586511e-16j
-3.25471719e-17+8.81681133e-16j 6.16651591e-17+1.59015677e-16j
3.14635470e-16+2.12937307e-16j -1.18140254e-02+1.28552941e-03j
1.18140254e-02-1.28552941e-03j -1.80844922e-16+8.41340886e-17j
1.78615569e-17+2.70011768e-16j -4.65274673e-17+9.34849019e-16j
1.29367576e-16+1.97524062e-16j 2.64508169e-16+3.24349044e-16j
4.82911320e-16+5.76082182e-16j 6.27969898e-16-4.72712147e-17j
6.81219495e-03+3.32206124e-03j -6.81219495e-03-3.32206124e-03j
7.38992201e-16-3.16803875e-16j -1.39311762e-16-3.23453896e-16j
5.11433383e-16+1.12182132e-16j -4.72098495e-17+1.51703847e-16j
2.51195966e-16-1.93511765e-16j -1.12757026e-17-1.14925430e-16j
5.35393878e-03-1.66115198e-03j -5.35393878e-03+1.66115198e-03j
4.81819445e-16+1.53956708e-16j 1.95598105e-16-3.36710327e-17j
9.04506532e-17+1.25506524e-17j 1.63631090e-16+1.55247123e-16j
6.91157282e-16+1.26216929e-16j -3.56291129e-16+5.00484060e-16j
4.91557507e-16+6.71710244e-17j -4.64753718e-16-1.30394279e-16j
-1.16895634e-16-4.46616720e-18j 2.44015932e-16-1.82750591e-16j
-7.60001168e-17+2.25361320e-16j 5.22151766e-16+3.44559450e-16j
6.81219495e-03+3.32206124e-03j -6.81219495e-03-3.32206124e-03j
1.33573708e-16+6.41630846e-16j -2.94469310e-16-2.01227923e-16j
-2.48235535e-03-7.62830643e-03j 2.48235535e-03+7.62830643e-03j
1.46584134e-16-5.63785130e-17j -4.65731334e-17-1.08286819e-16j
1.82031844e-16+1.03653535e-16j 6.03103098e-18+3.67393821e-16j
-6.67119402e-17-3.54115563e-16j -2.58698978e-16+2.47121582e-17j
-3.76167004e-16+4.55472952e-16j 6.00887640e-16+3.73918371e-16j
-2.15105711e-16+1.77809156e-16j 1.95943013e-02+2.45828327e-03j
-1.95943013e-02-2.45828327e-03j 2.50667542e-16+1.77158635e-16j]
<O_kin> = (-0.001676203291806102-0.0023445400116498074j)
<O_pot> = (3.917733396220584-0.663732907448769j)
<H> = (3.916057192928778-0.666077447460419j)
QDM Lattice 6x2#
[20]:
coup_j, coup_rk = (1, 1)
basis, model = setup_dimer_model(lattice_shape=(6, 2), n_solution=76, coup_j=coup_j, coup_rk=coup_rk)
evals, evecs = np.linalg.eigh(model.hamiltonian)
2023-12-23 21:48:24 [deep_first_search.py] INFO: Deep First Search starts.
2023-12-23 21:48:24 [deep_first_search.py] INFO: Found 76 Solutions as required in 693 steps.
[21]:
df = basis.dataframe.copy(deep=True)
df.index = format_custom_index(df.index)
df
[21]:
| 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | ... | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| (0) 4475835 | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | ... | 1 | 1 | 1 | 0 | 1 | 1 | 1 | 0 | 1 | 1 |
| (1) 4496307 | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 0 | ... | 1 | 1 | 1 | 0 | 1 | 1 | 0 | 0 | 1 | 1 |
| (2) 4508598 | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | ... | 1 | 1 | 1 | 0 | 1 | 1 | 0 | 1 | 1 | 0 |
| (3) 4774695 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 1 | ... | 1 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 1 |
| (4) 4803387 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 0 | 1 | ... | 1 | 1 | 0 | 0 | 1 | 1 | 1 | 0 | 1 | 1 |
| (5) 4823859 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | ... | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 |
| (6) 4836150 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 1 | ... | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 1 | 1 | 0 |
| (7) 4905747 | 0 | 1 | 0 | 0 | 1 | 0 | 1 | 0 | 1 | 1 | ... | 1 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 1 |
| (8) 5000043 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 1 | ... | 1 | 1 | 0 | 1 | 1 | 0 | 1 | 0 | 1 | 1 |
| (9) 5020515 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 0 | ... | 1 | 1 | 0 | 1 | 1 | 0 | 0 | 0 | 1 | 1 |
| (10) 5032806 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | ... | 1 | 1 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 |
| (11) 5081970 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | 1 | 1 | 0 | ... | 1 | 1 | 0 | 1 | 1 | 1 | 0 | 0 | 1 | 0 |
| (12) 5163858 | 0 | 1 | 0 | 0 | 1 | 1 | 1 | 0 | 1 | 1 | ... | 1 | 1 | 0 | 1 | 0 | 1 | 0 | 0 | 1 | 0 |
| (13) 8704695 | 1 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | ... | 1 | 0 | 1 | 0 | 1 | 1 | 0 | 1 | 1 | 1 |
| (14) 9032247 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 1 | ... | 1 | 0 | 0 | 0 | 1 | 1 | 0 | 1 | 1 | 1 |
| (15) 9228903 | 1 | 0 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | ... | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 1 |
| (16) 9257595 | 1 | 0 | 0 | 0 | 1 | 1 | 0 | 1 | 0 | 1 | ... | 1 | 0 | 0 | 1 | 1 | 1 | 1 | 0 | 1 | 1 |
| (17) 9278067 | 1 | 0 | 0 | 0 | 1 | 1 | 0 | 1 | 1 | 0 | ... | 1 | 0 | 0 | 1 | 1 | 1 | 0 | 0 | 1 | 1 |
| (18) 9290358 | 1 | 0 | 0 | 0 | 1 | 1 | 0 | 1 | 1 | 1 | ... | 1 | 0 | 0 | 1 | 1 | 1 | 0 | 1 | 1 | 0 |
| (19) 9359955 | 1 | 0 | 0 | 0 | 1 | 1 | 1 | 0 | 1 | 1 | ... | 1 | 0 | 0 | 1 | 0 | 1 | 0 | 0 | 1 | 1 |
| (20) 9716667 | 1 | 0 | 0 | 1 | 0 | 1 | 0 | 0 | 0 | 1 | ... | 1 | 1 | 1 | 0 | 1 | 1 | 1 | 0 | 1 | 1 |
| (21) 9737139 | 1 | 0 | 0 | 1 | 0 | 1 | 0 | 0 | 1 | 0 | ... | 1 | 1 | 1 | 0 | 1 | 1 | 0 | 0 | 1 | 1 |
| (22) 9749430 | 1 | 0 | 0 | 1 | 0 | 1 | 0 | 0 | 1 | 1 | ... | 1 | 1 | 1 | 0 | 1 | 1 | 0 | 1 | 1 | 0 |
| (23) 10015527 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 1 | 1 | ... | 1 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 1 |
| (24) 10044219 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 0 | 1 | ... | 1 | 1 | 0 | 0 | 1 | 1 | 1 | 0 | 1 | 1 |
| (25) 10064691 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | ... | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 |
| (26) 10076982 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 1 | ... | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 1 | 1 | 0 |
| (27) 10146579 | 1 | 0 | 0 | 1 | 1 | 0 | 1 | 0 | 1 | 1 | ... | 1 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 1 |
| (28) 10240875 | 1 | 0 | 0 | 1 | 1 | 1 | 0 | 0 | 0 | 1 | ... | 1 | 1 | 0 | 1 | 1 | 0 | 1 | 0 | 1 | 1 |
| (29) 10261347 | 1 | 0 | 0 | 1 | 1 | 1 | 0 | 0 | 1 | 0 | ... | 1 | 1 | 0 | 1 | 1 | 0 | 0 | 0 | 1 | 1 |
| (30) 10273638 | 1 | 0 | 0 | 1 | 1 | 1 | 0 | 0 | 1 | 1 | ... | 1 | 1 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 |
| (31) 10322802 | 1 | 0 | 0 | 1 | 1 | 1 | 0 | 1 | 1 | 0 | ... | 1 | 1 | 0 | 1 | 1 | 1 | 0 | 0 | 1 | 0 |
| (32) 10404690 | 1 | 0 | 0 | 1 | 1 | 1 | 1 | 0 | 1 | 1 | ... | 1 | 1 | 0 | 1 | 0 | 1 | 0 | 0 | 1 | 0 |
| (33) 11325735 | 1 | 0 | 1 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | ... | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 1 |
| (34) 11354427 | 1 | 0 | 1 | 0 | 1 | 1 | 0 | 1 | 0 | 1 | ... | 0 | 1 | 0 | 0 | 1 | 1 | 1 | 0 | 1 | 1 |
| (35) 11374899 | 1 | 0 | 1 | 0 | 1 | 1 | 0 | 1 | 1 | 0 | ... | 0 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 |
| (36) 11387190 | 1 | 0 | 1 | 0 | 1 | 1 | 0 | 1 | 1 | 1 | ... | 0 | 1 | 0 | 0 | 1 | 1 | 0 | 1 | 1 | 0 |
| (37) 11456787 | 1 | 0 | 1 | 0 | 1 | 1 | 1 | 0 | 1 | 1 | ... | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 1 |
| (38) 12863163 | 1 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | ... | 1 | 0 | 1 | 0 | 1 | 1 | 1 | 0 | 1 | 1 |
| (39) 12883635 | 1 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 0 | ... | 1 | 0 | 1 | 0 | 1 | 1 | 0 | 0 | 1 | 1 |
| (40) 12895926 | 1 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | ... | 1 | 0 | 1 | 0 | 1 | 1 | 0 | 1 | 1 | 0 |
| (41) 12903093 | 1 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | ... | 1 | 0 | 1 | 0 | 1 | 1 | 0 | 1 | 0 | 1 |
| (42) 13162023 | 1 | 1 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 1 | ... | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 1 |
| (43) 13190715 | 1 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 0 | 1 | ... | 1 | 0 | 0 | 0 | 1 | 1 | 1 | 0 | 1 | 1 |
| (44) 13211187 | 1 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | ... | 1 | 0 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 |
| (45) 13223478 | 1 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 1 | ... | 1 | 0 | 0 | 0 | 1 | 1 | 0 | 1 | 1 | 0 |
| (46) 13230645 | 1 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 1 | ... | 1 | 0 | 0 | 0 | 1 | 1 | 0 | 1 | 0 | 1 |
| (47) 13293075 | 1 | 1 | 0 | 0 | 1 | 0 | 1 | 0 | 1 | 1 | ... | 1 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 1 |
| (48) 13387371 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 1 | ... | 1 | 0 | 0 | 1 | 1 | 0 | 1 | 0 | 1 | 1 |
| (49) 13407843 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 0 | ... | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 1 | 1 |
| (50) 13420134 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | ... | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 |
| (51) 13427301 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | ... | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 0 | 1 |
| (52) 13469298 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 1 | 1 | 0 | ... | 1 | 0 | 0 | 1 | 1 | 1 | 0 | 0 | 1 | 0 |
| (53) 13476465 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 1 | 1 | 0 | ... | 1 | 0 | 0 | 1 | 1 | 1 | 0 | 0 | 0 | 1 |
| (54) 13551186 | 1 | 1 | 0 | 0 | 1 | 1 | 1 | 0 | 1 | 1 | ... | 1 | 0 | 0 | 1 | 0 | 1 | 0 | 0 | 1 | 0 |
| (55) 13558353 | 1 | 1 | 0 | 0 | 1 | 1 | 1 | 0 | 1 | 1 | ... | 1 | 0 | 0 | 1 | 0 | 1 | 0 | 0 | 0 | 1 |
| (56) 13928370 | 1 | 1 | 0 | 1 | 0 | 1 | 0 | 0 | 1 | 0 | ... | 1 | 1 | 1 | 0 | 1 | 1 | 0 | 0 | 1 | 0 |
| (57) 13935537 | 1 | 1 | 0 | 1 | 0 | 1 | 0 | 0 | 1 | 0 | ... | 1 | 1 | 1 | 0 | 1 | 1 | 0 | 0 | 0 | 1 |
| (58) 14173995 | 1 | 1 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | ... | 1 | 1 | 0 | 0 | 1 | 0 | 1 | 0 | 1 | 1 |
| (59) 14194467 | 1 | 1 | 0 | 1 | 1 | 0 | 0 | 0 | 1 | 0 | ... | 1 | 1 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 1 |
| (60) 14206758 | 1 | 1 | 0 | 1 | 1 | 0 | 0 | 0 | 1 | 1 | ... | 1 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 |
| (61) 14213925 | 1 | 1 | 0 | 1 | 1 | 0 | 0 | 0 | 1 | 1 | ... | 1 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 0 | 1 |
| (62) 14255922 | 1 | 1 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | ... | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 0 |
| (63) 14263089 | 1 | 1 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | ... | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 1 |
| (64) 14337810 | 1 | 1 | 0 | 1 | 1 | 0 | 1 | 0 | 1 | 1 | ... | 1 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 0 |
| (65) 14344977 | 1 | 1 | 0 | 1 | 1 | 0 | 1 | 0 | 1 | 1 | ... | 1 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 |
| (66) 14452578 | 1 | 1 | 0 | 1 | 1 | 1 | 0 | 0 | 1 | 0 | ... | 1 | 1 | 0 | 1 | 1 | 0 | 0 | 0 | 1 | 0 |
| (67) 14459745 | 1 | 1 | 0 | 1 | 1 | 1 | 0 | 0 | 1 | 0 | ... | 1 | 1 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 1 |
| (68) 15484203 | 1 | 1 | 1 | 0 | 1 | 1 | 0 | 0 | 0 | 1 | ... | 0 | 1 | 0 | 0 | 1 | 0 | 1 | 0 | 1 | 1 |
| (69) 15504675 | 1 | 1 | 1 | 0 | 1 | 1 | 0 | 0 | 1 | 0 | ... | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 1 |
| (70) 15516966 | 1 | 1 | 1 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | ... | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 |
| (71) 15524133 | 1 | 1 | 1 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | ... | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 0 | 1 |
| (72) 15566130 | 1 | 1 | 1 | 0 | 1 | 1 | 0 | 1 | 1 | 0 | ... | 0 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 0 |
| (73) 15573297 | 1 | 1 | 1 | 0 | 1 | 1 | 0 | 1 | 1 | 0 | ... | 0 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 1 |
| (74) 15648018 | 1 | 1 | 1 | 0 | 1 | 1 | 1 | 0 | 1 | 1 | ... | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 0 |
| (75) 15655185 | 1 | 1 | 1 | 0 | 1 | 1 | 1 | 0 | 1 | 1 | ... | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 |
76 rows × 24 columns
[23]:
evecs_df = pd.DataFrame.from_dict(
{
"eval": evals,
"kin": [(evec.T @ model.kinetic_term @ evec).item() for evec in evecs.T],
"pot": [(evec.T @ model.potential_term @ evec).item() for evec in evecs.T],
}
)
evecs_df[evecs_df["kin"].abs() < 1e-12]
[23]:
| eval | kin | pot | |
|---|---|---|---|
| 34 | 4.0 | -2.081668e-16 | 4.0 |
| 36 | 4.0 | -3.712308e-16 | 4.0 |
| 37 | 4.0 | 1.013079e-15 | 4.0 |
| 38 | 4.0 | 4.718448e-16 | 4.0 |
| 39 | 4.0 | -6.383782e-16 | 4.0 |
| 40 | 4.0 | 7.979728e-16 | 4.0 |
| 41 | 4.0 | 1.483189e-16 | 4.0 |
[22]:
k_rank = np.linalg.matrix_rank(model.kinetic_term)
h_rank = np.linalg.matrix_rank(model.hamiltonian)
print(k_rank, h_rank, basis.n_states, (basis.n_states - k_rank) / basis.n_states)
60 75 76 0.21052631578947367
[6]:
fig, axes = plt.subplots(19, 4, figsize=(30, 60), facecolor="white")
ax = axes.flatten()
for i in range(basis.n_states):
g = GraphVisualizer(SquareLattice(*model.shape, basis.dataframe.iloc[i]))
g.plot(show=False, ax=ax[i], node_size=800)
ax[i].set_title(f"{df.index[i]}")
[24]:
g = nx.from_numpy_array(-model.kinetic_term)
# left, right = nx.bipartite.sets(g)
# pos = nx.bipartite_layout(g, left, align='horizontal')
# pos = nx.spectral_layout(g)
# pos = {k: v + np.random.normal(0, 0.03, 2) for k, v in pos.items()}
# fig = plt.figure(figsize=(82, 32))
# nx.draw(g, pos, with_labels=True, node_size=5000, font_size=42, arrows=True, connectionstyle="arc3,rad=0.15")
# plt.show()
plt.spy(evecs, precision=1e-12, markersize=1)
plt.show()
[95]:
net = nt.Network(notebook=True, filter_menu=True, select_menu=True, font_color="black")
net.from_nx(g)
for node in net.nodes:
node["title"] = str(node["id"])
net.show_buttons(filter_=['nodes', 'edges', 'physics'])
net.show("nx.html")
Warning: When cdn_resources is 'local' jupyter notebook has issues displaying graphics on chrome/safari. Use cdn_resources='in_line' or cdn_resources='remote' if you have issues viewing graphics in a notebook.
nx.html
[95]:
[81]:
ig = igraph.Graph.from_networkx(g)
fig, ax = plt.subplots(figsize=(10, 8), facecolor="white")
igraph.plot(ig, layout=ig.layout_kamada_kawai(), vertex_size=10, vertex_color="orange", target=ax)
[81]:
<igraph.drawing.matplotlib.graph.GraphArtist at 0x13210e350>
[121]:
entropies = np.asarray([model.entropy(evecs[:, i], 0, 0) for i in range(basis.n_states)])
entropy_df = pd.Series(entropies, name="entropy")
entropy_df.loc[(entropy_df.index >= 20) & (entropy_df.index <= 50) & (entropy_df < 1.75)]
[121]:
20 1.058436
21 1.059513
22 1.061746
23 1.054004
24 1.074146
25 1.075756
26 1.012101
27 1.045869
28 1.041082
29 1.036557
30 0.997521
31 1.003358
32 1.054331
33 1.045045
34 1.032524
35 1.043826
36 1.053413
37 1.049003
38 1.061464
39 1.071072
40 1.050495
41 1.034524
42 1.093099
43 1.087584
44 0.949660
45 1.050145
46 1.028086
47 1.045876
48 1.041796
49 1.025918
50 1.020980
Name: entropy, dtype: float64
[122]:
entropy_df.plot(style='o')
[122]:
<Axes: >
[27]:
i = 41
plt.plot(evecs[:, i], linestyle='--', marker='o')
np.where(np.abs(evecs[:, i]) > 0.15)[0]
[27]:
array([ 7, 12, 19, 23, 29, 30, 33, 44, 49, 52, 53, 57, 60, 61, 62, 64, 68,
72])
[106]:
evecs_df.iloc[44]
[106]:
eval 4.678363
kin 0.730499
pot 3.947864
Name: 44, dtype: float64
[130]:
coup_j = np.ones((np.prod(model.shape), 1))
coup_j[0] = 1.05
energy_lump_model = QuantumLinkModel(coup_j, coup_rk, model.shape, basis)
energy_lump_ham = energy_lump_model.hamiltonian
# scar_state = np.zeros((basis.n_states, 1))
# scar_state[np.array([30, 45, 49, 26, 29, 44, 15, 52, 60, 17, 23, 62]), 0] = np.array([-1, -1, -1, 1, 1, 1, 1, 1, 1, -1, -1, -1]) # based on 44
# scar_state[np.array([3, 11, 16, 32, 36, 47, 58, 69]), 0] = np.array([1, -1, -1, 1, 1, -1, 1, -1]) # based on 24
# scar_state /= np.linalg.norm(scar_state)
scar_state = evecs[:, 44]
start, stop, num = 0, 2000, 100
evol_states = expm_multiply(-1j * energy_lump_ham, scar_state, start=start, stop=stop, num=num)
fidelity = [np.abs(evol_state.T @ scar_state).item() ** 2 for evol_state in evol_states]
entropy = [model.entropy(evol_state.flatten(), 1, 0) for evol_state in evol_states]
plt.plot(np.linspace(start, stop, num), fidelity, linestyle="--", marker="o")
plt.ylim(0.0001, 1.1)
# plt.yscale('log')
plt.xlabel("t")
plt.ylabel("Fidelity(t)")
plt.tight_layout()
plt.show()
plt.plot(np.linspace(start, stop, num), entropy, linestyle="--", marker="o")
plt.xlabel("t")
plt.ylabel("EE(t)")
plt.tight_layout()
plt.show()
final_state = np.real_if_close(evol_states[-1], tol=1e-12)
print(f"final fidelity = {fidelity[-1]}")
print(f"O_kin @ psi_scar = {(model.kinetic_term @ final_state).flatten()}")
print(f"<O_kin> = {(final_state.T @ model.kinetic_term @ final_state).item()}")
print(f"<O_pot> = {(final_state.T @ model.potential_term @ final_state).item()}")
print(f"<H> = {(final_state.T @ model.hamiltonian @ final_state).item()}")
final fidelity = 0.9684200052672003
O_kin @ psi_scar = [ 4.74315894e-03-0.01351011j 1.03124779e-02+0.0318116j
-7.33306781e-03-0.03792991j 1.52997953e-02+0.18422271j
1.03124779e-02+0.0318116j 5.65523991e-03-0.00780859j
-2.38554269e-03+0.01385663j -3.13183015e-03+0.0050453j
-7.33306781e-03-0.03792991j -2.38554269e-03+0.01385663j
-8.56180912e-03-0.00084907j -1.06443298e-02-0.21186007j
3.77536184e-03+0.01009899j -6.65967834e-02-0.23316644j
4.68541366e-03+0.00990252j 4.80413953e-02+0.15468343j
8.00044595e-02+0.21580224j -5.40763198e-02-0.1522819j
-1.46383257e-02-0.00309953j 2.79780403e-03-0.00045922j
8.00265490e-03+0.03862601j -4.03437097e-03+0.01099935j
-8.74394704e-03+0.02450193j -6.58516552e-03-0.12468163j
-4.03437097e-03+0.01099935j -5.63272425e-04+0.00214595j
4.15657095e-02+0.13874161j -8.01670083e-03+0.00049375j
-8.74394704e-03+0.02450193j 4.15657095e-02+0.13874161j
-2.04103940e-02-0.15438129j -1.65784189e-02-0.00708621j
4.32299982e-02+0.22165957j -1.01782856e-02+0.00430125j
-5.06235089e-03+0.00535867j 3.50300113e-03-0.0122706j
-5.43469476e-02-0.20895467j -4.47406122e-03-0.03199298j
-1.39877506e-02-0.03288221j -4.09086539e-03-0.01194989j
-1.99571287e-03-0.01061221j -7.85315053e-03+0.0032013j
1.65833918e-02+0.01981576j -4.09086539e-03-0.01194989j
3.82538196e-02+0.14726209j -3.61579908e-02-0.15076817j
5.70365967e-02+0.21604356j -4.83700054e-02-0.2317743j
-1.99571287e-03-0.01061221j -3.61579908e-02-0.15076817j
-4.72334768e-03+0.00378277j 5.41275120e-05+0.00747382j
4.61661124e-03+0.1427896j 1.67261583e-02-0.01445576j
3.83624394e-03-0.00639957j 3.94113378e-03+0.03493185j
8.00044595e-02+0.21580224j -5.06235089e-03+0.00535867j
-6.65967834e-02-0.23316644j 4.68541366e-03+0.00990252j
4.80413953e-02+0.15468343j -1.01782856e-02+0.00430125j
-5.40763198e-02-0.1522819j 3.50300113e-03-0.0122706j
2.79780403e-03-0.00045922j -4.47406122e-03-0.03199298j
-1.46383257e-02-0.00309953j -5.43469476e-02-0.20895467j
-7.85315053e-03+0.0032013j 5.70365967e-02+0.21604356j
5.41275120e-05+0.00747382j 1.99522153e-02+0.04243186j
1.67261583e-02-0.01445576j -1.21787411e-02-0.03798336j
3.94113378e-03+0.03493185j -1.92233570e-03+0.01589628j]
<O_kin> = (-0.6240604520813793+0.3289334174913119j)
<O_pot> = (-3.4166034229539792+1.73700287895565j)
<H> = (-4.040663875035359+2.065936296446962j)
QDM Lattice 6x4#
[28]:
coup_j, coup_rk = (1, 1)
basis, model = setup_dimer_model(lattice_shape=(6, 4), n_solution=1456, coup_j=coup_j, coup_rk=coup_rk)
evals, evecs = np.linalg.eigh(model.hamiltonian)
2023-12-23 21:51:23 [deep_first_search.py] INFO: Deep First Search starts.
2023-12-23 21:51:32 [deep_first_search.py] INFO: Found 1456 Solutions as required in 27921 steps.
[29]:
df = basis.dataframe.copy(deep=True)
df.index = format_custom_index(df.index)
df
[29]:
| 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | ... | 38 | 39 | 40 | 41 | 42 | 43 | 44 | 45 | 46 | 47 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| (0) 75046264433595 | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | ... | 1 | 1 | 1 | 0 | 1 | 1 | 1 | 0 | 1 | 1 |
| (1) 75046599846843 | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | ... | 1 | 1 | 1 | 0 | 1 | 1 | 1 | 0 | 1 | 1 |
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1456 rows × 48 columns
[30]:
evecs_df = pd.DataFrame.from_dict(
{
"eval": evals,
"kin": [(evec.T @ model.kinetic_term @ evec).item() for evec in evecs.T],
"pot": [(evec.T @ model.potential_term @ evec).item() for evec in evecs.T],
}
)
evecs_df[evecs_df["kin"].abs() < 1e-12]
[30]:
| eval | kin | pot |
|---|
[ ]:
k_rank = np.linalg.matrix_rank(model.kinetic_term)
h_rank = np.linalg.matrix_rank(model.hamiltonian)
print(k_rank, h_rank, basis.n_states, (basis.n_states - k_rank) / basis.n_states)
[32]:
g = nx.from_numpy_array(-model.kinetic_term)
# left, right = nx.bipartite.sets(g)
# pos = nx.bipartite_layout(g, left, align='horizontal')
# pos = nx.spectral_layout(g)
# pos = {k: v + np.random.normal(0, 0.08, 2) for k, v in pos.items()}
# fig = plt.figure(figsize=(82, 32))
# nx.draw(g, pos, with_labels=True, node_size=5000, font_size=42, arrows=True, connectionstyle="arc3,rad=0.15")
# plt.show()
plt.spy(evecs, precision=1e-12, markersize=0.01)
plt.show()
[53]:
net = nt.Network(notebook=True, filter_menu=True, select_menu=True, font_color="black")
net.from_nx(g)
for node in net.nodes:
node["title"] = str(node["id"])
net.show_buttons(filter_=['nodes', 'edges', 'physics'])
net.show("nx.html")
Warning: When cdn_resources is 'local' jupyter notebook has issues displaying graphics on chrome/safari. Use cdn_resources='in_line' or cdn_resources='remote' if you have issues viewing graphics in a notebook.
nx.html
[53]:
[58]:
ig = igraph.Graph.from_networkx(g)
ig.vs["label"] = [str(i) for i in range(ig.vcount())]
color = ["orange" for i in range(ig.vcount())]
highlight = [
[1099, 1102, 1239, 1256],
[604, 601, 331, 311],
[830, 891, 205, 117],
[355, 868, 808, 352],
[390, 367, 655, 695],
[1047, 1212, 1226, 1082]
]
highlight_color = ["aqua", "deepskyblue", "yellowgreen", "royalblue", "blueviolet", "cornflowerblue"]
for i, nodes in enumerate(highlight):
for node in nodes:
color[node] = highlight_color[i]
ig.vs["color"] = color
# fig, ax = plt.subplots(figsize=(10, 10), facecolor="white")
igraph.plot(
ig,
layout=ig.layout_kamada_kawai(),
vertex_size=8,
# vertex_color="orange",
edge_width=0.4,
# edge_color="darkgray",
vertex_label_size=6,
# target=ax
)
[58]:
[136]:
entropies = np.asarray([model.entropy(evecs[:, i], 0, 0) for i in range(basis.n_states)])
entropy_df = pd.Series(entropies, name="entropy")
entropy_df[entropy_df < 1.6]
[136]:
254 0.768489
1219 1.511229
Name: entropy, dtype: float64
[137]:
entropy_df.plot(style='o')
[137]:
<Axes: >
[139]:
plt.plot(evecs[:, 254], linestyle='--', marker='o')
np.where(np.abs(evecs[:, 254]) > 0.15)[0]
[139]:
array([ 367, 390, 655, 695, 1099, 1102, 1239, 1256])
[62]:
fig, axes = plt.subplots(2, 2, figsize=(16, 12), facecolor="white")
ax = axes.flatten()
for i, val in enumerate([891, 829, 1213, 354]): #[1099, 1102, 1239, 1256]
g = GraphVisualizer(SquareLattice(*model.shape, basis.dataframe.iloc[val]))
g.plot(show=False, ax=ax[i], node_size=800)
ax[i].set_title(f"{df.index[val]}")
[113]:
g = GraphVisualizer(SquareLattice(*model.shape, basis.dataframe.iloc[1213]))
g.plot(node_size=800)
[42]:
coup_j = np.ones((np.prod(model.shape), 1))
coup_j[0] = 1.1
energy_lump_model = QuantumLinkModel(coup_j, coup_rk, model.shape, basis)
energy_lump_ham = energy_lump_model.hamiltonian
scar_state = np.zeros((basis.n_states, 1))
scar_state[np.array([1099, 1102, 1239, 1256]), 0] = np.array([1, -1, -1, 1])
scar_state /= np.linalg.norm(scar_state)
start, stop, num = 0, 200, 100
evol_states = expm_multiply(-1j * energy_lump_ham, scar_state, start=start, stop=stop, num=num)
fidelity = [np.abs(evol_state.T @ scar_state).item() ** 2 for evol_state in evol_states]
entropy = [model.entropy(evol_state.flatten(), 1, 0) for evol_state in evol_states]
plt.plot(np.linspace(start, stop, num), fidelity, linestyle="--", marker="o")
plt.ylim(0.0001, 1.1)
# plt.yscale('log')
plt.xlabel("t")
plt.ylabel("Fidelity(t)")
plt.tight_layout()
plt.show()
plt.plot(np.linspace(start, stop, num), entropy, linestyle="--", marker="o")
plt.xlabel("t")
plt.ylabel("EE(t)")
plt.tight_layout()
plt.show()
final_state = np.real_if_close(evol_states[-1], tol=1e-12)
print(f"final fidelity = {fidelity[-1]}")
print(f"O_kin @ psi_scar = {(model.kinetic_term @ final_state).flatten()}")
print(f"<O_kin> = {(final_state.T @ model.kinetic_term @ final_state).item()}")
print(f"<O_pot> = {(final_state.T @ model.potential_term @ final_state).item()}")
print(f"<H> = {(final_state.T @ model.hamiltonian @ final_state).item()}")
/Users/tandaolin/projects/qlinks/.venv/lib/python3.11/site-packages/scipy/sparse/linalg/_eigen/arpack/arpack.py:1272: RuntimeWarning: k >= N - 1 for N * N square matrix. Attempting to use scipy.linalg.eig instead.
warnings.warn("k >= N - 1 for N * N square matrix. "
final fidelity = 1.000000000000004
O_kin @ psi_scar = [0.+0.j 0.+0.j 0.+0.j ... 0.+0.j 0.+0.j 0.+0.j]
<O_kin> = 0j
<O_pot> = (-2.3934538551791857+3.204899162708247j)
<H> = (-2.3934538551791857+3.204899162708247j)
QDM Lattice 8x4#
[94]:
coup_j, coup_rk = (1, 1) # dfs 2 mins 50 secs
basis, model = setup_dimer_model(lattice_shape=(8, 4), n_solution=17412, coup_j=coup_j, coup_rk=coup_rk)
2023-12-23 22:58:43 [deep_first_search.py] INFO: Deep First Search starts.
2023-12-23 23:01:16 [deep_first_search.py] INFO: Found 17412 Solutions as required in 479085 steps.
[104]:
# basis.dataframe.to_parquet("qdm_8x4_lattice.parquet", index=False)
basis = ComputationBasis.from_parquet("qdm_8x4_lattice.parquet")
model = QuantumLinkModel(coup_j, coup_rk, (8, 4), basis)
[106]:
evals, evecs = eigsh(csr_matrix(model.hamiltonian), k=1000, which="SA")
---------------------------------------------------------------------------
KeyboardInterrupt Traceback (most recent call last)
Cell In[106], line 1
----> 1 evals, evecs = eigsh(csr_matrix(model.hamiltonian), k=1000, which="SA")
File ~/projects/qlinks/.venv/lib/python3.11/site-packages/scipy/sparse/linalg/_eigen/arpack/arpack.py:1697, in eigsh(A, k, M, sigma, which, v0, ncv, maxiter, tol, return_eigenvectors, Minv, OPinv, mode)
1695 with _ARPACK_LOCK:
1696 while not params.converged:
-> 1697 params.iterate()
1699 return params.extract(return_eigenvectors)
File ~/projects/qlinks/.venv/lib/python3.11/site-packages/scipy/sparse/linalg/_eigen/arpack/arpack.py:537, in _SymmetricArpackParams.iterate(self)
535 def iterate(self):
536 self.ido, self.tol, self.resid, self.v, self.iparam, self.ipntr, self.info = \
--> 537 self._arpack_solver(self.ido, self.bmat, self.which, self.k,
538 self.tol, self.resid, self.v, self.iparam,
539 self.ipntr, self.workd, self.workl, self.info)
541 xslice = slice(self.ipntr[0] - 1, self.ipntr[0] - 1 + self.n)
542 yslice = slice(self.ipntr[1] - 1, self.ipntr[1] - 1 + self.n)
KeyboardInterrupt:
[97]:
df = basis.dataframe.copy(deep=True)
df.index = format_custom_index(df.index)
df
[97]:
| 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | ... | 54 | 55 | 56 | 57 | 58 | 59 | 60 | 61 | 62 | 63 | |
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| (1895) 5303467071482245905 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | ... | 1 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 |
| (1896) 5303467071501906225 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | ... | 0 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 1 |
| (1897) 5303467071503216913 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | ... | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 |
| (1898) 5303467071811228593 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | ... | 1 | 1 | 1 | 0 | 1 | 1 | 0 | 0 | 0 | 1 |
| (1899) 5303467071816471345 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | ... | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 1 |
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| (1901) 5303467071837442353 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | ... | 0 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 1 |
| (1902) 5303467071838753041 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | ... | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 |
| (1903) 5304153174969334710 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | ... | 1 | 1 | 1 | 0 | 1 | 1 | 0 | 1 | 1 | 0 |
| (1904) 5304153174974577462 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | ... | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 1 | 1 | 0 |
| (1905) 5304153175053218742 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | ... | 1 | 1 | 1 | 0 | 1 | 1 | 0 | 1 | 1 | 0 |
| (1906) 5304153175058461494 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | ... | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 1 | 1 | 0 |
| (1907) 5304153175079432502 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | ... | 0 | 1 | 0 | 0 | 1 | 1 | 0 | 1 | 1 | 0 |
| (1908) 5304223548008348598 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | ... | 1 | 1 | 1 | 0 | 1 | 1 | 0 | 1 | 1 | 0 |
| (1909) 5304223548013591350 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | ... | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 1 | 1 | 0 |
| (1910) 5304223548092232630 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | ... | 1 | 1 | 1 | 0 | 1 | 1 | 0 | 1 | 1 | 0 |
| (1911) 5304223548097475382 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | ... | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 1 | 1 | 0 |
| (1912) 5304223548118446390 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | ... | 0 | 1 | 0 | 0 | 1 | 1 | 0 | 1 | 1 | 0 |
| (1913) 5304223569482660790 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | ... | 1 | 1 | 1 | 0 | 1 | 1 | 0 | 1 | 1 | 0 |
| (1914) 5304223569487903542 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | ... | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 1 | 1 | 0 |
| (1915) 5304223569566544822 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | ... | 1 | 1 | 1 | 0 | 1 | 1 | 0 | 1 | 1 | 0 |
| (1916) 5304223569571787574 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | ... | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 1 | 1 | 0 |
| (1917) 5304223569592758582 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | ... | 0 | 1 | 0 | 0 | 1 | 1 | 0 | 1 | 1 | 0 |
| (1918) 5304311506791087030 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | ... | 1 | 1 | 1 | 0 | 1 | 1 | 0 | 1 | 1 | 0 |
| (1919) 5304311506796329782 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | ... | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 1 | 1 | 0 |
| (1920) 5304311506874971062 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | ... | 1 | 1 | 1 | 0 | 1 | 1 | 0 | 1 | 1 | 0 |
| (1921) 5304311506880213814 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | ... | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 1 | 1 | 0 |
| (1922) 5304311506901184822 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | ... | 0 | 1 | 0 | 0 | 1 | 1 | 0 | 1 | 1 | 0 |
| (1923) 5304311508133231542 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | ... | 1 | 1 | 1 | 0 | 1 | 1 | 0 | 1 | 1 | 0 |
| (1924) 5304311508138474294 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | ... | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 1 | 1 | 0 |
| (1925) 5304311508217115574 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | ... | 1 | 1 | 1 | 0 | 1 | 1 | 0 | 1 | 1 | 0 |
| (1926) 5304311508222358326 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | ... | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 1 | 1 | 0 |
| (1927) 5304311508243329334 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | ... | 0 | 1 | 0 | 0 | 1 | 1 | 0 | 1 | 1 | 0 |
| (1928) 5304311508552651702 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | ... | 1 | 1 | 1 | 0 | 1 | 1 | 0 | 1 | 1 | 0 |
| (1929) 5304311508557894454 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | ... | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 1 | 1 | 0 |
| (1930) 5304311508578865462 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | ... | 0 | 1 | 0 | 0 | 1 | 1 | 0 | 1 | 1 | 0 |
| (1931) 5304311528265399222 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | ... | 1 | 1 | 1 | 0 | 1 | 1 | 0 | 1 | 1 | 0 |
| (1932) 5304311528270641974 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | ... | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 1 | 1 | 0 |
| (1933) 5304311528349283254 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | ... | 1 | 1 | 1 | 0 | 1 | 1 | 0 | 1 | 1 | 0 |
| (1934) 5304311528354526006 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | ... | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 1 | 1 | 0 |
| (1935) 5304311528375497014 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | ... | 0 | 1 | 0 | 0 | 1 | 1 | 0 | 1 | 1 | 0 |
| (1936) 5304311529607543734 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | ... | 1 | 1 | 1 | 0 | 1 | 1 | 0 | 1 | 1 | 0 |
| (1937) 5304311529612786486 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | ... | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 1 | 1 | 0 |
| (1938) 5304311529691427766 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | ... | 1 | 1 | 1 | 0 | 1 | 1 | 0 | 1 | 1 | 0 |
| (1939) 5304311529696670518 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | ... | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 1 | 1 | 0 |
| (1940) 5304311529717641526 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | ... | 0 | 1 | 0 | 0 | 1 | 1 | 0 | 1 | 1 | 0 |
| (1941) 5304311530026963894 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | ... | 1 | 1 | 1 | 0 | 1 | 1 | 0 | 1 | 1 | 0 |
| (1942) 5304311530032206646 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | ... | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 1 | 1 | 0 |
| (1943) 5304311530053177654 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | ... | 0 | 1 | 0 | 0 | 1 | 1 | 0 | 1 | 1 | 0 |
| (1944) 5309008472288050107 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | ... | 1 | 1 | 1 | 0 | 1 | 1 | 1 | 0 | 1 | 1 |
| (1945) 5309008472288377779 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | ... | 1 | 1 | 1 | 0 | 1 | 1 | 0 | 0 | 1 | 1 |
| (1946) 5309008472293292859 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | ... | 1 | 1 | 0 | 0 | 1 | 1 | 1 | 0 | 1 | 1 |
| (1947) 5309008472293620531 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | ... | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 |
| (1948) 5309008472294931219 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | ... | 1 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 1 |
| (1949) 5309008472371934139 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | ... | 1 | 1 | 1 | 0 | 1 | 1 | 1 | 0 | 1 | 1 |
| (1950) 5309008472372261811 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | ... | 1 | 1 | 1 | 0 | 1 | 1 | 0 | 0 | 1 | 1 |
| (1951) 5309008472377176891 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | ... | 1 | 1 | 0 | 0 | 1 | 1 | 1 | 0 | 1 | 1 |
| (1952) 5309008472377504563 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | ... | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 |
| (1953) 5309008472378815251 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | ... | 1 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 1 |
| (1954) 5309008472398147899 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | ... | 0 | 1 | 0 | 0 | 1 | 1 | 1 | 0 | 1 | 1 |
| (1955) 5309008472398475571 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | ... | 0 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 |
| (1956) 5309008472399786259 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | ... | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 1 |
| (1957) 5309008558192179987 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | ... | 1 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 1 |
| (1958) 5309008558276064019 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | ... | 1 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 1 |
| (1959) 5309008558297035027 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | ... | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 1 |
| (1960) 5309096431070788539 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | ... | 1 | 1 | 1 | 0 | 1 | 1 | 1 | 0 | 1 | 1 |
| (1961) 5309096431071116211 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | ... | 1 | 1 | 1 | 0 | 1 | 1 | 0 | 0 | 1 | 1 |
| (1962) 5309096431076031291 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | ... | 1 | 1 | 0 | 0 | 1 | 1 | 1 | 0 | 1 | 1 |
| (1963) 5309096431076358963 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | ... | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 |
| (1964) 5309096431077669651 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | ... | 1 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 1 |
| (1965) 5309096431154672571 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | ... | 1 | 1 | 1 | 0 | 1 | 1 | 1 | 0 | 1 | 1 |
| (1966) 5309096431155000243 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | ... | 1 | 1 | 1 | 0 | 1 | 1 | 0 | 0 | 1 | 1 |
| (1967) 5309096431159915323 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | ... | 1 | 1 | 0 | 0 | 1 | 1 | 1 | 0 | 1 | 1 |
| (1968) 5309096431160242995 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | ... | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 |
| (1969) 5309096431161553683 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | ... | 1 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 1 |
| (1970) 5309096431180886331 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | ... | 0 | 1 | 0 | 0 | 1 | 1 | 1 | 0 | 1 | 1 |
| (1971) 5309096431181214003 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | ... | 0 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 |
| (1972) 5309096431182524691 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | ... | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 1 |
| (1973) 5309096432412933051 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | ... | 1 | 1 | 1 | 0 | 1 | 1 | 1 | 0 | 1 | 1 |
| (1974) 5309096432413260723 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | ... | 1 | 1 | 1 | 0 | 1 | 1 | 0 | 0 | 1 | 1 |
| (1975) 5309096432418175803 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | ... | 1 | 1 | 0 | 0 | 1 | 1 | 1 | 0 | 1 | 1 |
| (1976) 5309096432418503475 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | ... | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 |
| (1977) 5309096432419814163 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | ... | 1 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 1 |
| (1978) 5309096432496817083 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | ... | 1 | 1 | 1 | 0 | 1 | 1 | 1 | 0 | 1 | 1 |
| (1979) 5309096432497144755 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | ... | 1 | 1 | 1 | 0 | 1 | 1 | 0 | 0 | 1 | 1 |
| (1980) 5309096432502059835 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | ... | 1 | 1 | 0 | 0 | 1 | 1 | 1 | 0 | 1 | 1 |
| (1981) 5309096432502387507 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | ... | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 |
| (1982) 5309096432503698195 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | ... | 1 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 1 |
| (1983) 5309096432523030843 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | ... | 0 | 1 | 0 | 0 | 1 | 1 | 1 | 0 | 1 | 1 |
| (1984) 5309096432523358515 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | ... | 0 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 |
| (1985) 5309096432524669203 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | ... | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 1 |
| (1986) 5309096432832353211 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | ... | 1 | 1 | 1 | 0 | 1 | 1 | 1 | 0 | 1 | 1 |
| (1987) 5309096432832680883 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | ... | 1 | 1 | 1 | 0 | 1 | 1 | 0 | 0 | 1 | 1 |
| (1988) 5309096432837595963 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | ... | 1 | 1 | 0 | 0 | 1 | 1 | 1 | 0 | 1 | 1 |
| (1989) 5309096432837923635 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | ... | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 |
| (1990) 5309096432839234323 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | ... | 1 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 1 |
| (1991) 5309096432858566971 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | ... | 0 | 1 | 0 | 0 | 1 | 1 | 1 | 0 | 1 | 1 |
| (1992) 5309096432858894643 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | ... | 0 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 |
| (1993) 5309096432860205331 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | ... | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 1 |
| (1994) 5309096433487068081 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | ... | 1 | 1 | 1 | 0 | 1 | 1 | 0 | 0 | 0 | 1 |
| (1995) 5309096433492310833 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | ... | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 1 |
| (1996) 5309096433493621521 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | ... | 1 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 |
| (1997) 5309096433570952113 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | ... | 1 | 1 | 1 | 0 | 1 | 1 | 0 | 0 | 0 | 1 |
| (1998) 5309096433576194865 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | ... | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 1 |
| (1999) 5309096433577505553 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | ... | 1 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 |
| (2000) 5309096433597165873 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | ... | 0 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 1 |
| (2001) 5309096433598476561 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | ... | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 |
| (2002) 5309096433906488241 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | ... | 1 | 1 | 1 | 0 | 1 | 1 | 0 | 0 | 0 | 1 |
| (2003) 5309096433911730993 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | ... | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 1 |
| (2004) 5309096433913041681 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | ... | 1 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 |
| (2005) 5309096433932702001 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | ... | 0 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 1 |
| (2006) 5309096433934012689 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | ... | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 |
| (2007) 5309096516974918419 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | ... | 1 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 1 |
| (2008) 5309096517058802451 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | ... | 1 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 1 |
| (2009) 5309096517079773459 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | ... | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 1 |
| (2010) 5309096518317062931 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | ... | 1 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 1 |
| (2011) 5309096518400946963 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | ... | 1 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 1 |
| (2012) 5309096518421917971 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | ... | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 1 |
| (2013) 5309096518736483091 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | ... | 1 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 1 |
| (2014) 5309096518757454099 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | ... | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 1 |
| (2015) 5309096519390870289 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | ... | 1 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 |
| (2016) 5309096519474754321 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | ... | 1 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 |
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17412 rows × 64 columns
[56]:
g = nx.from_numpy_array(-model.kinetic_term)
ig = igraph.Graph.from_networkx(g)
igraph.plot(
ig,
# layout=ig.layout_kamada_kawai(),
vertex_size=4,
vertex_color="orange",
edge_width=0.1,
edge_color="gray"
)
[56]:
[43]:
plt.spy(evecs, precision=1e-12, markersize=0.05)
plt.show()
---------------------------------------------------------------------------
KeyboardInterrupt Traceback (most recent call last)
Cell In[43], line 10
2 # left, right = nx.bipartite.sets(g)
3 # pos = nx.bipartite_layout(g, left, align='horizontal')
4 # pos = nx.spectral_layout(g)
(...)
7 # nx.draw(g, pos, with_labels=True, node_size=5000, font_size=42, arrows=True, connectionstyle="arc3,rad=0.15")
8 # plt.show()
9 ig = igraph.Graph.from_networkx(g)
---> 10 igraph.plot(ig, layout=ig.layout_kamada_kawai(), vertex_size=10, vertex_color="orange")
11 plt.show()
13 plt.spy(evecs, precision=1e-12, markersize=0.05)
File ~/projects/qlinks/.venv/lib/python3.11/site-packages/igraph/layout.py:691, in _layout_method_wrapper.<locals>.result(*args, **kwds)
690 def result(*args, **kwds):
--> 691 layout = func(*args, **kwds)
692 if not isinstance(layout, Layout):
693 layout = Layout(layout)
KeyboardInterrupt:
Check effective matrix#
[71]:
l = 0.0
dj = 1.1
mat = np.array(
[
[l, dj, dj, 0, 0, 0, 0, 0, 0],
[dj, l, 0, 1, 1, 0, 0, 0, 0],
[dj, 0, l, 0, 1, 1, 0, 0, 0],
[0, 1, 0, l, 0, 0, 1, 0, 0],
[0, 1, 1, 0, l, 0, 1, 1, 0],
[0, 0, 1, 0, 0, l, 0, 1, 0],
[0, 0, 0, 1, 1, 0, l, 0, 1],
[0, 0, 0, 0, 1, 1, 0, l, 1],
[0, 0, 0, 0, 0, 0, 1, 1, l]
]
)
np.linalg.matrix_power(mat, 2) @ np.array([[0, 1, -1, 0, 0, 0, -1, 1, 0]]).T
[71]:
array([[0.],
[0.],
[0.],
[0.],
[0.],
[0.],
[0.],
[0.],
[0.]])
[75]:
from IPython.display import clear_output
from time import sleep
for i in range(20):
print(f"matrix power {i+1}")
walker = np.linalg.matrix_power(mat, i+1)
walker /= np.linalg.norm(walker)
plt.matshow(walker, animated=True)
plt.colorbar()
plt.show()
sleep(0.3)
clear_output(wait=True)
matrix power 20
QLM Lattice 4x2#
[5]:
coup_j, coup_rk = (1, 1)
basis, model = setup_link_model(lattice_shape=(4, 2), n_solution=38, coup_j=coup_j, coup_rk=coup_rk)
evals, evecs = np.linalg.eigh(model.hamiltonian)
2023-12-23 23:44:19 [deep_first_search.py] INFO: Deep First Search starts.
2023-12-23 23:44:19 [deep_first_search.py] INFO: Found 38 Solutions as required in 231 steps.
[6]:
df = basis.dataframe.copy(deep=True)
df.index = format_custom_index(df.index)
df
[6]:
| 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| (0) 1455 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 1 | 1 | 0 | 1 | 0 | 1 | 1 | 1 | 1 |
| (1) 4539 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 1 | 0 | 1 | 1 | 1 | 0 | 1 | 1 |
| (2) 5310 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 0 | 1 | 0 | 1 | 1 | 1 | 1 | 1 | 0 |
| (3) 6582 | 0 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 1 | 1 | 0 | 1 | 1 | 0 |
| (4) 9627 | 0 | 0 | 1 | 0 | 0 | 1 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 1 | 1 |
| (5) 11670 | 0 | 0 | 1 | 0 | 1 | 1 | 0 | 1 | 1 | 0 | 0 | 1 | 0 | 1 | 1 | 0 |
| (6) 16875 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 0 | 1 | 0 | 1 | 1 |
| (7) 17646 | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 1 | 0 | 1 | 1 | 1 | 0 |
| (8) 18093 | 0 | 1 | 0 | 0 | 0 | 1 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 1 | 0 | 1 |
| (9) 18918 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 1 | 1 | 0 | 0 | 1 | 1 | 0 |
| (10) 19365 | 0 | 1 | 0 | 0 | 1 | 0 | 1 | 1 | 1 | 0 | 1 | 0 | 0 | 1 | 0 | 1 |
| (11) 20730 | 0 | 1 | 0 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 0 | 1 | 0 |
| (12) 21177 | 0 | 1 | 0 | 1 | 0 | 0 | 1 | 0 | 1 | 0 | 1 | 1 | 1 | 0 | 0 | 1 |
| (13) 23220 | 0 | 1 | 0 | 1 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 1 | 0 | 1 | 0 | 0 |
| (14) 25818 | 0 | 1 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | 1 | 1 | 0 | 1 | 0 |
| (15) 26265 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 |
| (16) 27090 | 0 | 1 | 1 | 0 | 1 | 0 | 0 | 1 | 1 | 1 | 0 | 1 | 0 | 0 | 1 | 0 |
| (17) 27537 | 0 | 1 | 1 | 0 | 1 | 0 | 1 | 1 | 1 | 0 | 0 | 1 | 0 | 0 | 0 | 1 |
| (18) 28308 | 0 | 1 | 1 | 0 | 1 | 1 | 1 | 0 | 1 | 0 | 0 | 1 | 0 | 1 | 0 | 0 |
| (19) 37227 | 1 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 1 | 1 | 0 | 1 | 0 | 1 | 1 |
| (20) 37998 | 1 | 0 | 0 | 1 | 0 | 1 | 0 | 0 | 0 | 1 | 1 | 0 | 1 | 1 | 1 | 0 |
| (21) 38445 | 1 | 0 | 0 | 1 | 0 | 1 | 1 | 0 | 0 | 0 | 1 | 0 | 1 | 1 | 0 | 1 |
| (22) 39270 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 |
| (23) 39717 | 1 | 0 | 0 | 1 | 1 | 0 | 1 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 0 | 1 |
| (24) 42315 | 1 | 0 | 1 | 0 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 0 | 1 | 0 | 1 | 1 |
| (25) 44358 | 1 | 0 | 1 | 0 | 1 | 1 | 0 | 1 | 0 | 1 | 0 | 0 | 0 | 1 | 1 | 0 |
| (26) 44805 | 1 | 0 | 1 | 0 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 1 |
| (27) 46170 | 1 | 0 | 1 | 1 | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 1 | 1 | 0 | 1 | 0 |
| (28) 46617 | 1 | 0 | 1 | 1 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 0 | 0 | 1 |
| (29) 47442 | 1 | 0 | 1 | 1 | 1 | 0 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 0 | 1 | 0 |
| (30) 47889 | 1 | 0 | 1 | 1 | 1 | 0 | 1 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 |
| (31) 48660 | 1 | 0 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 0 |
| (32) 53865 | 1 | 1 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | 1 | 0 | 0 | 1 |
| (33) 55908 | 1 | 1 | 0 | 1 | 1 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 0 | 0 |
| (34) 58953 | 1 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 |
| (35) 60225 | 1 | 1 | 1 | 0 | 1 | 0 | 1 | 1 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 1 |
| (36) 60996 | 1 | 1 | 1 | 0 | 1 | 1 | 1 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 0 |
| (37) 64080 | 1 | 1 | 1 | 1 | 1 | 0 | 1 | 0 | 0 | 1 | 0 | 1 | 0 | 0 | 0 | 0 |
[35]:
evecs_df = pd.DataFrame.from_dict(
{
"eval": evals,
"kin": [(evec.T @ model.kinetic_term @ evec).item() for evec in evecs.T],
"pot": [(evec.T @ model.potential_term @ evec).item() for evec in evecs.T],
}
)
evecs_df[evecs_df["kin"].abs() < 1e-12]
[35]:
| eval | kin | pot | |
|---|---|---|---|
| 16 | 4.0 | -6.434147e-30 | 4.0 |
| 18 | 4.0 | -4.659210e-30 | 4.0 |
| 19 | 4.0 | -5.482583e-29 | 4.0 |
| 20 | 4.0 | -1.774937e-30 | 4.0 |
| 21 | 5.0 | 4.240127e-30 | 5.0 |
| 22 | 5.0 | 3.056836e-29 | 5.0 |
| 23 | 5.0 | 2.465190e-30 | 5.0 |
| 24 | 5.0 | 6.064368e-30 | 5.0 |
[36]:
k_rank = np.linalg.matrix_rank(model.kinetic_term)
h_rank = np.linalg.matrix_rank(model.hamiltonian)
print(k_rank, h_rank, basis.n_states, (basis.n_states - k_rank) / basis.n_states)
24 37 38 0.3684210526315789
[43]:
plt.spy(evecs, precision=1e-12, markersize=5.0)
plt.show()
[46]:
g = nx.from_numpy_array(-model.kinetic_term)
ig = igraph.Graph.from_networkx(g)
ig.vs["label"] = [str(i) for i in range(ig.vcount())]
igraph.plot(
ig,
layout=ig.layout_kamada_kawai(),
vertex_size=16,
vertex_color="orange",
)
[46]:
[52]:
entropies = np.asarray([model.entropy(evecs[:, i], 1, 0) for i in range(basis.n_states)])
entropy_df = pd.Series(entropies, name="entropy")
entropy_df[entropy_df < 1.8]
[52]:
7 1.656653
8 1.641885
18 1.795919
19 1.696326
29 1.655564
36 1.342927
37 1.576712
Name: entropy, dtype: float64
[51]:
entropy_df.plot(style='o')
[51]:
<Axes: >
[53]:
i = 16
plt.plot(evecs[:, i], linestyle='--', marker='o')
np.where(np.abs(evecs[:, i]) > 0.15)[0]
[53]:
array([ 1, 10, 13, 16, 21, 24, 27, 36])
[57]:
fig, axes = plt.subplots(10, 4, figsize=(18, 40), facecolor="white")
ax = axes.flatten()
for i in range(basis.n_states):
g = GraphVisualizer(SquareLattice(*model.shape, basis.dataframe.iloc[i]))
g.plot(show=False, ax=ax[i], node_size=800)
ax[i].set_title(f"{df.index[i]}")
[8]:
coup_j = np.ones((np.prod(model.shape), 1))
coup_j[1] = 1.5
energy_lump_model = QuantumLinkModel(coup_j, coup_rk, model.shape, basis)
energy_lump_ham = energy_lump_model.hamiltonian
scar_state = np.zeros((basis.n_states, 1))
scar_state[np.array([1, 13, 24, 36]), 0] = np.array([-1, 1, 1, -1])
scar_state /= np.linalg.norm(scar_state)
start, stop, num = 0, 1000, 100
evol_states = expm_multiply(-1j * energy_lump_ham, scar_state, start=start, stop=stop, num=num)
fidelity = [np.abs(evol_state.T @ scar_state).item() ** 2 for evol_state in evol_states]
entropy = [model.entropy(evol_state.flatten(), 1, 0) for evol_state in evol_states]
plt.plot(np.linspace(start, stop, num), fidelity, linestyle="--", marker="o")
plt.ylim(0.0001, 1.1)
# plt.yscale('log')
plt.xlabel("t")
plt.ylabel("Fidelity(t)")
plt.tight_layout()
plt.show()
plt.plot(np.linspace(start, stop, num), np.round(entropy, 12), linestyle="--", marker="o")
plt.xlabel("t")
plt.ylabel("EE(t)")
plt.tight_layout()
plt.show()
final_state = np.real_if_close(evol_states[-1], tol=1e-12)
print(f"final fidelity = {fidelity[-1]}")
print(f"O_kin @ psi_scar = {(model.kinetic_term @ final_state).flatten()}")
print(f"<O_kin> = {(final_state.T @ model.kinetic_term @ final_state).item()}")
print(f"<O_pot> = {(final_state.T @ model.potential_term @ final_state).item()}")
print(f"<H> = {(final_state.T @ model.hamiltonian @ final_state).item()}")
/Users/tandaolin/projects/qlinks/.venv/lib/python3.11/site-packages/scipy/sparse/linalg/_eigen/arpack/arpack.py:1272: RuntimeWarning: k >= N - 1 for N * N square matrix. Attempting to use scipy.linalg.eig instead.
warnings.warn("k >= N - 1 for N * N square matrix. "
final fidelity = 0.7741120945201118
O_kin @ psi_scar = [-1.26816175e-01+2.16458990e-01j -1.74860126e-15-3.71924713e-15j
-2.49510615e-15+3.02725097e-17j 4.06474110e-16+1.59762660e-15j
6.27588676e-02-6.25774945e-02j -1.26737070e-15-7.74890865e-16j
1.26816175e-01-2.16458990e-01j -1.28369537e-15+8.18789481e-16j
6.27588676e-02-6.25774945e-02j -6.27588676e-02+6.25774945e-02j
-5.72458747e-16-9.02056208e-16j -1.88632146e-15-8.22702315e-16j
8.30786076e-16+4.32674238e-17j -2.58126853e-15-5.27355937e-16j
-1.53222678e-15+4.74340519e-17j -1.83878645e-15-1.16591307e-15j
-2.35823835e-15+2.43020098e-16j 8.02828449e-16-6.46296253e-16j
-6.27588676e-02+6.25774945e-02j -6.27588676e-02+6.25774945e-02j
-1.28905152e-15+2.05400241e-15j -2.00000525e-15+1.21800850e-15j
-4.23278892e-15+8.57837956e-16j 1.06105148e-16+8.60137442e-16j
8.32667268e-16-1.22124533e-15j -2.63677968e-15+5.55111512e-17j
-9.56214122e-16+3.54749084e-16j -3.78516662e-15+2.35922393e-15j
-6.27588676e-02+6.25774945e-02j 6.27588676e-02-6.25774945e-02j
-3.07393000e-15+6.52256027e-16j 1.26816175e-01-2.16458990e-01j
-4.89783893e-16-9.79329937e-16j 6.27588676e-02-6.25774945e-02j
-1.24900090e-15-9.43689571e-16j -1.42226886e-16-3.88756955e-16j
0.00000000e+00+1.97064587e-15j -1.26816175e-01+2.16458990e-01j]
<O_kin> = (-0.04469297354403623-0.17216447669592444j)
<O_pot> = (-2.3013209894209448-2.8002538832799444j)
<H> = (-2.3460139629649808-2.972418359975869j)
QLM Lattice 4x4#
[59]:
coup_j, coup_rk = (1, 1)
basis, model = setup_link_model(lattice_shape=(4, 4), n_solution=990, coup_j=coup_j, coup_rk=coup_rk)
evals, evecs = np.linalg.eigh(model.hamiltonian)
2023-12-23 22:09:14 [deep_first_search.py] INFO: Deep First Search starts.
2023-12-23 22:09:19 [deep_first_search.py] INFO: Found 990 Solutions as required in 9381 steps.
[60]:
df = basis.dataframe.copy(deep=True)
df.index = format_custom_index(df.index)
df
[60]:
| 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | ... | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 30 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| (0) 84258735 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 0 | ... | 1 | 1 | 1 | 0 | 1 | 0 | 1 | 1 | 1 | 1 |
| (1) 88518063 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 1 | ... | 0 | 1 | 1 | 0 | 1 | 0 | 1 | 1 | 1 | 1 |
| (2) 88534701 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 1 | ... | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 1 | 0 | 1 |
| (3) 88843695 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 1 | ... | 0 | 1 | 1 | 0 | 1 | 0 | 1 | 1 | 1 | 1 |
| (4) 88860333 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 1 | ... | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 1 | 0 | 1 |
| (5) 88861605 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 1 | ... | 1 | 1 | 1 | 0 | 1 | 0 | 0 | 1 | 0 | 1 |
| (6) 93728175 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 1 | 1 | 0 | ... | 0 | 1 | 1 | 0 | 1 | 0 | 1 | 1 | 1 | 1 |
| (7) 93744813 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 1 | 1 | 0 | ... | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 1 | 0 | 1 |
| (8) 93765165 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 1 | 1 | 0 | ... | 1 | 0 | 0 | 0 | 1 | 0 | 1 | 1 | 0 | 1 |
| (9) 94053807 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 1 | 1 | 0 | ... | 0 | 1 | 1 | 0 | 1 | 0 | 1 | 1 | 1 | 1 |
| (10) 94070445 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 1 | 1 | 0 | ... | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 1 | 0 | 1 |
| (11) 94071717 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 1 | 1 | 0 | ... | 1 | 1 | 1 | 0 | 1 | 0 | 0 | 1 | 0 | 1 |
| (12) 94090797 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 1 | 1 | 0 | ... | 1 | 0 | 0 | 0 | 1 | 0 | 1 | 1 | 0 | 1 |
| (13) 94092069 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 1 | 1 | 0 | ... | 1 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 0 | 1 |
| (14) 95356335 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 1 | 1 | 0 | ... | 0 | 1 | 1 | 0 | 1 | 0 | 1 | 1 | 1 | 1 |
| (15) 95372973 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 1 | 1 | 0 | ... | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 1 | 0 | 1 |
| (16) 95374245 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 1 | 1 | 0 | ... | 1 | 1 | 1 | 0 | 1 | 0 | 0 | 1 | 0 | 1 |
| (17) 95393325 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 1 | 1 | 0 | ... | 1 | 0 | 0 | 0 | 1 | 0 | 1 | 1 | 0 | 1 |
| (18) 95394597 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 1 | 1 | 0 | ... | 1 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 0 | 1 |
| (19) 95399685 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 1 | 1 | 0 | ... | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 1 |
| (20) 286374843 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | ... | 1 | 1 | 1 | 0 | 1 | 1 | 1 | 0 | 1 | 1 |
| (21) 287677371 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | ... | 1 | 1 | 1 | 0 | 1 | 1 | 1 | 0 | 1 | 1 |
| (22) 287682459 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | ... | 1 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 1 | 1 |
| (23) 290634171 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 1 | ... | 0 | 1 | 1 | 0 | 1 | 1 | 1 | 0 | 1 | 1 |
| (24) 290650809 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 1 | ... | 1 | 0 | 1 | 0 | 1 | 1 | 1 | 0 | 0 | 1 |
| (25) 291936699 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 1 | ... | 0 | 1 | 1 | 0 | 1 | 1 | 1 | 0 | 1 | 1 |
| (26) 291941787 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 1 | ... | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 1 | 1 |
| (27) 291953337 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 1 | ... | 1 | 0 | 1 | 0 | 1 | 1 | 1 | 0 | 0 | 1 |
| (28) 291958425 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 1 | ... | 1 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 |
| (29) 292262331 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 1 | ... | 0 | 1 | 1 | 0 | 1 | 1 | 1 | 0 | 1 | 1 |
| (30) 292267419 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 1 | ... | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 1 | 1 |
| (31) 292278969 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 1 | ... | 1 | 0 | 1 | 0 | 1 | 1 | 1 | 0 | 0 | 1 |
| (32) 292284057 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 1 | ... | 1 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 |
| (33) 292285329 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 1 | ... | 1 | 1 | 1 | 0 | 0 | 1 | 0 | 0 | 0 | 1 |
| (34) 297146811 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 1 | 0 | ... | 0 | 1 | 1 | 0 | 1 | 1 | 1 | 0 | 1 | 1 |
| (35) 297151899 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 1 | 0 | ... | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 1 | 1 |
| (36) 297163449 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 1 | 0 | ... | 1 | 0 | 1 | 0 | 1 | 1 | 1 | 0 | 0 | 1 |
| (37) 297168537 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 1 | 0 | ... | 1 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 |
| (38) 297188889 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 1 | 0 | ... | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 0 | 0 | 1 |
| (39) 297472443 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 1 | 0 | ... | 0 | 1 | 1 | 0 | 1 | 1 | 1 | 0 | 1 | 1 |
| (40) 297477531 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 1 | 0 | ... | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 1 | 1 |
| (41) 297489081 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 1 | 0 | ... | 1 | 0 | 1 | 0 | 1 | 1 | 1 | 0 | 0 | 1 |
| (42) 297494169 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 1 | 0 | ... | 1 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 |
| (43) 297495441 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 1 | 0 | ... | 1 | 1 | 1 | 0 | 0 | 1 | 0 | 0 | 0 | 1 |
| (44) 297514521 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 1 | 0 | ... | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 0 | 0 | 1 |
| (45) 297515793 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 1 | 0 | ... | 1 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 |
| (46) 336903870 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 0 | 0 | 0 | ... | 1 | 0 | 1 | 0 | 1 | 1 | 1 | 1 | 1 | 0 |
| (47) 337229502 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 0 | 0 | 0 | ... | 1 | 0 | 1 | 0 | 1 | 1 | 1 | 1 | 1 | 0 |
| (48) 337230774 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 0 | 0 | 0 | ... | 1 | 1 | 1 | 0 | 1 | 1 | 0 | 1 | 1 | 0 |
| (49) 338532030 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 0 | 0 | 0 | ... | 1 | 0 | 1 | 0 | 1 | 1 | 1 | 1 | 1 | 0 |
| (50) 338533302 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 0 | 0 | 0 | ... | 1 | 1 | 1 | 0 | 1 | 1 | 0 | 1 | 1 | 0 |
| (51) 338538390 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 0 | 0 | 0 | ... | 1 | 1 | 1 | 0 | 0 | 1 | 0 | 1 | 1 | 0 |
| (52) 341488830 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 0 | 0 | 1 | ... | 0 | 0 | 1 | 0 | 1 | 1 | 1 | 1 | 1 | 0 |
| (53) 341490102 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 0 | 0 | 1 | ... | 0 | 1 | 1 | 0 | 1 | 1 | 0 | 1 | 1 | 0 |
| (54) 341506740 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 0 | 0 | 1 | ... | 1 | 0 | 1 | 0 | 1 | 1 | 0 | 1 | 0 | 0 |
| (55) 342791358 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 0 | 0 | 1 | ... | 0 | 0 | 1 | 0 | 1 | 1 | 1 | 1 | 1 | 0 |
| (56) 342792630 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 0 | 0 | 1 | ... | 0 | 1 | 1 | 0 | 1 | 1 | 0 | 1 | 1 | 0 |
| (57) 342797718 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 0 | 0 | 1 | ... | 0 | 1 | 1 | 0 | 0 | 1 | 0 | 1 | 1 | 0 |
| (58) 342809268 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 0 | 0 | 1 | ... | 1 | 0 | 1 | 0 | 1 | 1 | 0 | 1 | 0 | 0 |
| (59) 342814356 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 0 | 0 | 1 | ... | 1 | 0 | 1 | 0 | 0 | 1 | 0 | 1 | 0 | 0 |
| (60) 348001470 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 0 | 1 | 0 | ... | 0 | 0 | 1 | 0 | 1 | 1 | 1 | 1 | 1 | 0 |
| (61) 348002742 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 0 | 1 | 0 | ... | 0 | 1 | 1 | 0 | 1 | 1 | 0 | 1 | 1 | 0 |
| (62) 348007830 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 0 | 1 | 0 | ... | 0 | 1 | 1 | 0 | 0 | 1 | 0 | 1 | 1 | 0 |
| (63) 348019380 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 0 | 1 | 0 | ... | 1 | 0 | 1 | 0 | 1 | 1 | 0 | 1 | 0 | 0 |
| (64) 348024468 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 0 | 1 | 0 | ... | 1 | 0 | 1 | 0 | 0 | 1 | 0 | 1 | 0 | 0 |
| (65) 348044820 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 0 | 1 | 0 | ... | 1 | 0 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 0 |
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990 rows × 32 columns
[61]:
evecs_df = pd.DataFrame.from_dict(
{
"eval": evals,
"kin": [(evec.T @ model.kinetic_term @ evec).item() for evec in evecs.T],
"pot": [(evec.T @ model.potential_term @ evec).item() for evec in evecs.T],
}
)
evecs_df[evecs_df["kin"].abs() < 1e-12]
[61]:
| eval | kin | pot | |
|---|---|---|---|
| 574 | 8.0 | -1.349265e-18 | 8.0 |
| 575 | 8.0 | 1.875301e-18 | 8.0 |
| 576 | 8.0 | 4.546683e-18 | 8.0 |
| 577 | 8.0 | 5.921708e-19 | 8.0 |
| 578 | 8.0 | 5.162581e-18 | 8.0 |
| 579 | 8.0 | -4.928153e-18 | 8.0 |
| 580 | 8.0 | -1.140181e-17 | 8.0 |
| 581 | 8.0 | -2.148948e-18 | 8.0 |
| 582 | 8.0 | -1.139690e-17 | 8.0 |
| 583 | 8.0 | 4.990670e-18 | 8.0 |
| 584 | 8.0 | 8.727366e-20 | 8.0 |
| 585 | 8.0 | -1.531264e-18 | 8.0 |
| 586 | 8.0 | 1.152132e-17 | 8.0 |
| 587 | 8.0 | -2.056418e-18 | 8.0 |
| 588 | 8.0 | 1.160960e-17 | 8.0 |
| 589 | 8.0 | 8.304220e-18 | 8.0 |
| 590 | 8.0 | -2.619799e-19 | 8.0 |
| 591 | 8.0 | 4.393195e-19 | 8.0 |
| 592 | 8.0 | -2.615988e-18 | 8.0 |
| 593 | 8.0 | -3.253685e-18 | 8.0 |
| 594 | 8.0 | 2.828691e-18 | 8.0 |
| 595 | 8.0 | -2.132213e-17 | 8.0 |
| 596 | 8.0 | -1.095384e-17 | 8.0 |
| 597 | 8.0 | -5.541874e-18 | 8.0 |
| 598 | 8.0 | -3.299282e-18 | 8.0 |
| 599 | 8.0 | 5.322860e-18 | 8.0 |
[62]:
k_rank = np.linalg.matrix_rank(model.kinetic_term)
h_rank = np.linalg.matrix_rank(model.hamiltonian)
print(k_rank, h_rank, basis.n_states, (basis.n_states - k_rank) / basis.n_states)
832 989 990 0.1595959595959596
[67]:
plt.spy(evecs, precision=1e-12, markersize=0.05)
plt.show()
[74]:
g = nx.from_numpy_array(-model.kinetic_term)
ig = igraph.Graph.from_networkx(g)
ig.vs["label"] = [str(i) for i in range(ig.vcount())]
igraph.plot(
ig,
layout=ig.layout_kamada_kawai(),
vertex_size=10,
vertex_color="orange",
edge_width=0.8,
# edge_color="gray"
vertex_label_size=6,
)
[74]:
[77]:
entropies = np.asarray([model.entropy(evecs[:, i], 1, 0) for i in range(basis.n_states)])
entropy_df = pd.Series(entropies, name="entropy")
entropy_df[entropy_df < 3.5]
[77]:
12 3.437326
13 3.354152
16 3.485520
17 3.295924
18 3.337060
20 3.121310
29 3.189914
30 3.121809
31 3.227343
32 3.121694
55 3.341194
577 2.857901
579 3.233935
584 3.482694
586 3.285896
767 3.459376
786 3.191232
789 3.288996
842 3.490871
868 3.460691
905 3.426521
951 3.131694
955 3.079945
958 3.378943
959 3.397993
960 3.295655
961 3.222521
962 3.351094
963 3.388660
964 2.999035
965 3.302850
966 3.402129
973 3.164994
974 3.376386
975 3.334039
976 3.404896
977 3.450523
978 3.294743
982 3.410129
988 2.059264
989 2.229480
Name: entropy, dtype: float64
[76]:
entropy_df.plot(style='o')
[76]:
<Axes: >
[89]:
i = 574
plt.plot(evecs[:, i], linestyle='--', marker='o')
np.where(np.abs(evecs[:, i]) > 0.09)[0]
[89]:
array([ 35, 82, 90, 91, 140, 183, 216, 226, 257, 299, 307, 331, 333,
334, 338, 378, 412, 455, 534, 577, 611, 651, 655, 656, 658, 682,
690, 732, 763, 773, 806, 849, 898, 899, 907, 954])
[87]:
fig, axes = plt.subplots(4, 2, figsize=(16, 24), facecolor="white")
ax = axes.flatten()
for i, val in enumerate([35, 82, 299, 333, 656, 690, 732, 773]):
g = GraphVisualizer(SquareLattice(*model.shape, basis.dataframe.iloc[val]))
g.plot(show=False, ax=ax[i], node_size=800)
ax[i].set_title(f"{df.index[val]}")
[90]:
coup_j = np.ones((np.prod(model.shape), 1))
# coup_j[1] = 1.1
energy_lump_model = QuantumLinkModel(coup_j, coup_rk, model.shape, basis)
energy_lump_ham = energy_lump_model.hamiltonian
scar_state = np.zeros((basis.n_states, 1))
scar_state[np.array([82, 333]), 0] = np.array([1, -1])
scar_state /= np.linalg.norm(scar_state)
start, stop, num = 0, 200, 100
evol_states = expm_multiply(-1j * energy_lump_ham, scar_state, start=start, stop=stop, num=num)
fidelity = [np.abs(evol_state.T @ scar_state).item() ** 2 for evol_state in evol_states]
entropy = [model.entropy(evol_state.flatten(), 2, 0) for evol_state in evol_states]
plt.plot(np.linspace(start, stop, num), fidelity, linestyle="--", marker="o")
plt.ylim(0.0001, 1.1)
# plt.yscale('log')
plt.xlabel("t")
plt.ylabel("Fidelity(t)")
plt.tight_layout()
plt.show()
plt.plot(np.linspace(start, stop, num), np.round(entropy, 12), linestyle="--", marker="o")
plt.xlabel("t")
plt.ylabel("EE(t)")
plt.tight_layout()
plt.show()
final_state = np.real_if_close(evol_states[0], tol=1e-12)
print(f"final fidelity = {fidelity[0]}")
print(f"O_kin @ psi_scar = {(model.kinetic_term @ final_state).flatten()}")
print(f"<O_kin> = {(final_state.T @ model.kinetic_term @ final_state).item()}")
print(f"<O_pot> = {(final_state.T @ model.potential_term @ final_state).item()}")
print(f"<H> = {(final_state.T @ model.hamiltonian @ final_state).item()}")
/Users/tandaolin/projects/qlinks/.venv/lib/python3.11/site-packages/scipy/sparse/linalg/_eigen/arpack/arpack.py:1272: RuntimeWarning: k >= N - 1 for N * N square matrix. Attempting to use scipy.linalg.eig instead.
warnings.warn("k >= N - 1 for N * N square matrix. "
final fidelity = 0.9999999999999996
O_kin @ psi_scar = [ 0. 0. 0. 0. 0. 0.
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<O_kin> = 0.0
<O_pot> = 7.999999999999998
<H> = 7.999999999999998
[ ]: